Modeling and Prediction of Pedestrian Behavior based on the Sub-goal Concept

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1 Modelng and Predcton of Pedestran Behavor based on the Sub-goal Concet Tetsush Ieda, Yoshhro Chgodo, Danel Rea, Francesco Zanlungo, Masahro Shom, Taayu Kanda Intellgent Robotcs and Communcaton Laboratores, ATR, Kyoto, Jaan E-mal: Abstract Ths study addresses a method to redct edestrans' long term behavor n order to enable a robot to rovde them servces. In order to do that we want to be able to redct ther fnal goal and the traectory they wll follow to reach t. We attan ths tas borrowng from human scence studes the concet of sub-goals, defned as onts and landmars of the envronment towards whch edestrans wal or where they tae drectonal choces before reachng the fnal destnaton. We retreve the oston of these sub-goals from the analyss of a large set of edestran traectores n a shong mall, and model ther global behavor through transton robabltes between sub-goals. The method allows us to redct the future oston of edestrans on the bass of the observaton of ther traectory u to the moment. Keywords-comonent; edestran models; sub-goal retreval; behavor antcaton I. INTRODUCTION A robot oeratng n an envronment where a sgnfcant number of edestrans s resent, such as a shong mall, needs a consderable amount of nformaton about the current and general behavor of edestrans n the envronment. For the tas of navgaton, besdes the obvous need to now the ostons and veloctes of nearby edestrans for local lannng, also the global lanner can tae advantage of redcton of human behavor. Knowledge of the current edestran densty at larger dstances, the average behavor of edestrans n the envronment, and even the use of a edestran smulator to redct the future behavor of the crowd as a whole may revent the robot to end n hghly oulated areas where collson avodance could not be erformed effectvely. To rovde servces such as gudng [,], survellance, shong assstance [3], gvng nformaton [4,5,6], assstng eole who get lost, an ablty to recognze behavoral atterns and redct future behavor and oston s needed. In ths wor we tacle the roblem of oston redcton, whch s of artcular mortance for a moble robot rovdng servces n a shong mall. An ablty of redctng the oston of edestrans beyond velocty roecton wll enhance the collson avodance of the robot [7], and enable aroachng humans to tal wth them and rovdng servces [8,9]. In artcular, we develo an aroach that gves us nformaton useful not only for redcton but also for robot navgaton, envronment modelng and edestran smulaton. To do that we use the concet of sub-goals,.e. we assume that edestrans have the tendency to segment the walng route to Ths research was suorted by JST, CREST. Fgure. Sub-goal: a ont n a sace edestrans are walng towards The subgoals n the cture are retreved wth our method, see fg. ther fnal destnaton n a olygonal lne assng through some onts nown as sub-goals [,] (Fg. ). We assume that the oston of these sub-goals s to a certan extent common to all edestrans,.e. that they are determned manly by the envronment. We retreve the oston of these sub-goals from the observaton of a large number of traectores, and buld a robablstc transton model between sub-goals that s used for moton redcton. The sub-goals could be used both as the nodes of the robot global ath lanner, and as the nodes of the lanner n the edestran smulator, ntroducng a common nowledge of the envronmental features based on the actual moton of edestrans. II. RELATED WORK The smlest method to redct the moton of a edestran s to erform a velocty-based lnear roecton,.e. assumng that eole ee movng wth a constant seed. Ths s nown to be a reasonable aroxmaton for short-term behavor, used for collson avodance [,3] and tracng algorthms [4]. The method does not need any nowledge about the envronment or the revous traectory, but t s not relable for long term redcton gnorng envronmental constrants or nfluence. In order to redct the future oston of a waler usng the nformaton based on the revously observed behavor of a large number of edestrans, the moton can be modeled through transtons among grds or nodes [5,6]. Recent aroaches often retreve tycal transton atterns (atternbased aroaches) to use also nowledge about the revous behavor of the edestran whose moton s beng redcted [7]. The future course of a new observed traectory s estmated wth transton atterns that resemble the new traectory [7]. Pattern-based methods use a dscrete sace descrton. Usually both for robot navgaton and edestran smulaton a contnuous descrton n whch the nodes of the global ath lanner are gven at arbtrary dstance s referred. The usual

2 Fgure. Model of a global behavor: a edestran follows a seres of subgoals whle avodng collson wth others way to do that n robotcs s to buld a toologcal ma from the geometrcal features of the envronment. These toologcal methods do not use nformaton comng from human behavor, and though effectve for navgaton, they do not rovde any nowledge about human behavor to the robot system [8]. Bennewtz et al. [7] use a sto ont aroach,.e. retreve the oston of the nodes based on observaton of human behavor, comutng them as the laces where eole usually sto. Ths aroach could be lmted n the descrton of an envronment n whch eole tae decsons or ust change drecton wthout stong. Once the grah s bult wth toologcal or humanmoton based aroach, the nodes are used as the states of a hdden Marov model, and the transton robabltes are used to model and redct human behavor. In ths wor n order to buld the nodes of the transton model we use the socal scence concet of sub-goal. The route of edestrans can be modeled as a successon of decson onts, whch are nfluenced by the structure of the envronment, as for examle by the resence of landmars and archtectural features [,,9]. By learnng from eole s traectores, we retreve the sub-goals from the drectons edestrans mostly head to. To reduce the effect of local dynamcs, ths drecton to the sub-goal s comuted subtractng the effect of local collson avodng as redcted by the Socal Force Model secfcaton ntroduced n []. III. OVERVIEW OF THE MODEL OF PEDESTRIAN BEHAVIOR Ths secton rovdes a short overvew of the model of edestran behavor we roose (fgure ). A. Fnal destnaton (goal) We assume that edestrans move towards a fnal destnaton, ther goal. In an deally unform envronment, wthout any nd of fxed or movng obstacle, nether dfferent archtectural features and the le, edestrans would move straght to ther goal. In a real envronment, the goal s usually not reachable on a straght ath due to the resence of fxed obstacles, and aths dfferent from the straght one could be referred due to easness of walng (nature of terran, archtectural features, crowdng). B. Influence of the envronment (sub-goals) We assume that n a real envronment edestrans try to dvde ther ath (global ath lannng) n straght segments onng ther actual oston wth the goal [], and we assume Fgure 3. Overvew of comutaton n the roosed method the nodes of these segments to be common to all edestrans,.e. to be determned only by the nature of the envronment. Snce the ath to the next node s straght, these nodes are called sub-goals. These sub-goals can be determned by the resence of a vsual landmar, and edestrans may swtch to ther new sub-goal when t s vsble to them, even before reachng the one they were walng to. C. Influence from other edestrans (local behavor) Devatons from the straght ath to next sub-goal are due to the collson avodng (local ath lannng) behavor. Ths nteracton can be exressed n a smle mathematcal form alyng the Socal Force Model [] (SFM), that exresses the acceleraton of edestrans as v v ( t) a( t ) f ( t) Here v, stands for the referred velocty of the edestran, whose magntude s determned by the most comfortable seed for the edestran, and the drecton by the unt vector ontng from the current oston to the next sub-goal; s the tme scale for the edestran to recover her referred velocty, whle f s the collson avodng behavor towards edestran (the force deends on the relatve ostons and veloctes of edestrans). Dfferent secfcatons of f have been roosed, and we adot the secfcaton and arameters roosed n [9], whch have shown to best descrbe edestran behavor n the SFM framewor. We use the SFM framewor because eq () can be easly arranged as v v ( t ) ( a( t) f ( t)) () n order to obtan v,.e. the drecton to the next goal, from edestran ostons, veloctes and acceleratons. IV. PREDICTION ALGORITHM A. Overvew Fgure 3 shows the archtecture of the roosed method, whch conssts of an offlne analyss ste and an onlne redcton ste. In the offlne analyss ste, from the ostons and veloctes of a large number of edestrans we comute the ()

3 referred veloctes usng eq. (). Then the sub-goals n the envronment are estmated by extractng onts toward whch many edestrans head. In the roosed model, each edestran s consdered to head towards one of the sub-goals and sometmes swtch to a dfferent one. So we reresent each traectory by a sequence of sub-goals, and obtan a robablstc sub-goal transton model from observed traectores. In the onlne ste, after observng the ntal orton of a traectory, we comute the referred velocty wth eq. (). On the bass of the referred velocty and sub-goal sequence of a edestran u to the moment, we estmate future ostons usng a robablstc transton model. B. Estmatng sub-goals After collectng a large amount of traectores n the envronment, and comutng the referred veloctes, we dvde the envronment n a dscrete grd and study the dstrbuton of referred drectons for each grd cell. We retreve sub-goals as onts towards whch a large number of drectons converge, and n order to fnd them we defne a sub-goal feld extendng the drecton robabltes from each grd cell and defnng the sub-goal set as the set of maxma of the sub-goal feld. ) Comutng Drectons of Pedestran Moton on a Grd We dscretze the envronment usng a square grd wth lnear dmenson.5 meters whch roughly corresonds to the sze of a human beng, and thus to the scale at whch we can exect behavoral varaton. For each grd the dstrbuton of referred drectons s obtaned from veloctes v.the dstrbuton of drectons at each grd square may consst of several dfferent comonents. Fgures 4 (a), (b) show the observed drecton dstrbuton n a grd square. We call each comonent flow, and we model ts angle dstrbuton usng a von Mses dstrbuton [3], ex( cos( )) (, ) I ( ) where s the mean, s the varance and I ( ) s the order modfed Bessel functon of frst nd [4]. Eq. (3) gves the aroxmaton of a Gaussan wraed on the crcle. Subsequently we estmate each comonent usng an EM algorthm. We frst dscretze and smooth the drecton dstrbuton on the grd cell, and then comute the number of local maxma of ths dstrbuton. We assume that to each local maxmum corresonds a flow, and fx the arameter to the oston of the local maxmum, and to the wdth of the ea around the maxmum. After ths ntalzaton rocess, each edestran drecton s assocated to one of the von Mses mxture comonents accordng to the robablty the comonent assgns to that drecton. Then each dstrbuton comonent s estmated based on the assgned edestran drectons. Ths rocess s reeated untl the estmated comonent has converged [5]. (3) Ths rocess defnes the set of flows F { },.e. the set of all the von Mses comonents found n the envronment, each one characterzed by ts and values. Fgure 4 (c) shows the estmated flows detected n an observed area. ) Estmatng Sub-goals from Moton Drectons By extendng flow drectons from each grd, we can defne a scalar sub-goal feld on the envronment. As shown n fgure 5 n the case of two grd cells, a maxmum of ths feld s a lace n the envronment towards whch many eole head,.e. a sub-goal. We defne the scalar feld generated by the flow n ont x as a) Observed traectores of edestrans that go across a grd square. b) The sold lne shows the hstogram of drectons θ, and the dashed lnes show each comonent n the estmated von Mses mxture dstrbuton. c) Estmated maor flow drectons the grd square. The drectons of the arrows corresond to the mean of the estmated von Mses dstrbuton n b). Fgure 4. Observed traectores and the estmated flow drectons n a grd square. Fgure 5. Extenton of flows from grd cells. (+) reresents a ont where two flows cross, whch s a oston many eole head toward and s a sub-goal canddate. Fgure 6. Contrbuton of a flow to the feld (x). ( x) (, ) ; arg( x c) (4)

4 sub-goal sequence at each tme nstant ( ): 3 smoothed sub-goal sequence: seres of sub-goals: Fgure 7. Estmatng the sub-goal that a edestran heads towards. where and are the arameters of the flow accordng to eq. (3). s the drecton of the dslacement vector from the grd center c, where the flow s defned, to x (Fgure 6), whle arg() returns the angle of the gven vector. The scalar feld generated by a sub-set of flows F F s defned as ( x ; ) ( x) (5) F s.t. F and, followng the examle wth two flows of fgure 6, a subgoal s generated by a flow set F s the maxmum of feld (5), s argmax ( x; F ). (6) x Equaton (6) defnes a set S { s } of sub goals gven a artton of flows { F }, wth F F and F F. At the same tme, we defne a flow artton from a sub-goal set sayng that F s argmax ( s )}. (7) { s S In the followng we assume N sub-goals to be resent n our envronment, and thus the set of flows F to be dvded n N subsets F. Our algorthm ales teratvely eqs. (6) and (7) n order to fnd the flow artton { F } and sub-goal set S that maxmze ss ( s ; F ) (8) The algorthm s ntalzed searchng for sub-goals on the dscrete grd flows are defned on. At the begnnng the flow artton conssts of the whole set, F F, and the frst subgoal s s the maxmum on the grd of ( x; F ). The second subgoal s s obtaned as the ont on the grd that, after obtanng the artton { F, F } from eq. (7), maxmzes eq (8),.e. ( s; F ) ( s; F ). The rocess s contnued untl N sub-goals and flow arttons are obtaned, and at ths ont, usng a local Monte Carlo search, eq. (6) s used to obtan the sub-goal ostons n contnuous sace, whch ends the ntalzaton rocess. Subsequently, eqs. (7) and (6) are aled untl convergence,.e. the total dslacement of sub-goals under alcaton of eq. (6) s smaller than a gven threshold. C. Modelng Global Behavor Gven a set of sub-goals, we model each edestran's traectory as a sequence of movements toward one of the subgoals. So we model the long-term behavor of edestrans by condtonal robablty dstrbuton of the transton to the next sub-goal. ) Comutng a seres of ast sub-goals from a fullyobserved traectory To comute the seres of sub-goals from a traectory, frst we estmate the sub-goal of the edestran at each nstant t based on the referred velocty. When the edestran s at oston x wth referred velocty v, the sub-goal wth closest angle to v that exsts n a vsual cone area C s selected ( y n Fgure 7) as s argmn(angle( v, y x )), (9) t y n C t where angle( x, y ) s a functon that returns the angle between vector x and y n [, ]. The area of the vsual cone s defned relatve to x and v and ts sze set to 4 degrees n angle, a value close to the tycal flow sread. Then we smooth the sub-goal sequence { s t } obtaned at the revous ste nto { s t } by selectng the most frequent ndex that aears n a fxed length tme wndow: st mode( st /,, st / ), () where mode( ) s a functon that returns most frequent value and s the length of the tme wndow, set as msec, a value emrcally chosen to smooth nose wthout losng nformaton. Fnally we comute the sub-goal seres whch s defned by the frst sub-goal and the sub-goals that are dfferent from the revous one (fg. 7): seres of sub-goals = s } { s s s }. () t { t t t ) Transton robablty To model common movements n an envronment, we construct a statstcal model of sub-goal transtons. We comute the followng transton robabltes based on observed traectores. # of ( y y ) = edestrans that go toward # of edestrans that y then y go toward y () We also comute transton model wth n- ste hstores (n-gram model) ( y N y,, y N # of )= edestrans that go toward # of edestrans that go toward y,, y y, y N N (3) We comuted these transtons for values of n u to 6, a comromse between comutatonal economy and comleteness of descrton of the envronment. D. Predcton of Future Movements After observng traectores of edestrans u to now, our algorthm redcts future traectory of each edestran.

5 ) Estmatng the robablty of the current sub-goal from a artally observed traectory From the observed traectores, the referred velocty vector v and sub-goal sequence untl now { q } are comuted for each edestran. Suose v v s the referred seed, and arg( v ) s the referred drecton. Based on Bayes' Theorem, we model the robablty that the edestran heads toward sub-goal y gven the referred velocty and the subgoal sequence as: ( y,{ q }) ( { q }, y) ( y { q }). (4) The frst term of the rght hand sde of the equaton s the robablty of the referred drecton gven { q }, y, whch s emrcally derved from a large amount of observed traectores. In ths aer, we aroxmate ths term assumng deendence on only the last sub-goal q m, and model the term usng a Gaussan dstrbuton, whose mean and varance deend on q m, y and are determned on emrcal data. The second term s from the robablstc sub-goal transton model. Eq. (4) s comuted for all sub-goals y n the vsual cone. ) Predcton of future ostons The robablty that the edestran reaches a sub-goal z n future va any other sub-goal y s gven by z,{ q }) ( z,{ q }, y ) ( y,{ q }). (5) ( In the followng we wll assume that the transton robablty from y to the next sub-goal s ndeendent from the current referred drecton, so that the frst term n the summaton s smlfed as ( z,{ q }, y ) ( z { q }, y ) (6) To comute ths term we sum over all the ossble routes from y to z, but snce our goal s to estmate the oston of a edestran at T seconds after the last observaton, we tae n consderaton only the routes that are comatble wth the length a edestran can wal n tme T. Exlctly we assume ( z { q }, y ) = { r }:subgoal sequence from y to z L :set of { r } s. t. dst ( x, y, r,, r ) v T and dst( x, y, r,, r, z) v T where dst( r, r, )s the dstance along the route ( r, r, ) x :current oston of the edestran v :referred velocty T :tme elased { r } L ( z,{ r } { q }, y ) where the robabltes n the summaton term are obtaned recursvely from eq. (4). Fnally, the robablty to reach subgoal z gven the observed sub-goal sequence and resent referred velocty, eq. (5), s obtaned summng over all ossble sub-goals y n the cone of vson (usng eq. 4) n n (7) Sum of the length of sold lnes = T v Fgure 8. Comutaton of the estmated oston at tme T. The lnes from x to z show the sequence of sub-goals wth the hghest robablty. Note that edestran changes sub-goals before reachng them. We use statstcs of the dstance from each sub-goal to change sub-goals. Number of edestrans 3 Length of observatons 7 hours Sze of observaton area 86 m Fgure 9. Exermental envronment Pctures of the envronment are vsble n fgures, 4. The future oston of the edestran at tme T after the observaton s estmated assumng that wth robablty (5), (.e. the one that sums over the whole set L n (7)) the edestran wll be located on the lne connectng the sub-goal successon { r } that assumes maxmum robablty n L, and more recsely, t wll be located between sub-goal r n and subgoal z at a dstance v T ( v s the referred seed at x ) from x on that lne, see Fgure 8. V. RESULTS Ths secton llustrates the model obtaned wth our method and comares the redcton results wth those of other methods. In secton V-B, to evaluate the locatons of nodes, we comared our method wth other node generaton methods by runnng the same redcton rocedure descrbed n VI-C and VI-D. Then n secton V-C, we further comared our method and dfferent redcton methods. A. Envronment and setu We collected traectores n a large shong mall (Fg. 9) n whch several restaurants and shos are resent. Snce t s located between another large buldng and a tran staton, t s a transton lace for many edestrans. The sze of the observaton area s 86 square meters and we observed 3 edestrans n 7 hours, of whch 85% were used for calbraton and 5% for testng. To trac the edestrans we used twenty LRFs (Houyo Automatc UTM-3LX) and aled a tracng algorthm based on shae-matchng at torso-level [6]. Ths area s larger than those nvestgated n revous lterature wors, and thus the redcton tas s harder.

6 B. Descrton of the envronment through the sub-goal ostons In ths secton we comare the roosed method wth other methods that rovde a descrton of the envronment based on a grah, as toology based [8] methods and sto ont [7] methods. The comarson s erformed from a qualtatve ont of vew, comarng the descrton of the envronment gven by the oston of the nodes, and from a quanttatve ont of vew, comarng the recson n the redcton of future ostons. The toology method s mlemented obtanng the ma of the envronment from the edestran movement data. The envronment s dscretzed usng a grd, and the grd cells are dvded between those on whch edestran data was recorded, and those wthout edestran data. After a smoothng rocess, the boundares between the two areas are used as the boundares of the ma of the envronment. In order to buld the ma a Vorono dagram method [7] s aled. Also to mlement the sto-ont method we dvde the envronment usng a dscrete grd, and after defnng the number of stoonts N, we have chec the ostons of the N cells where edestrans sto more often. After obtanng the oston of the nodes, n order to redct the future oston of the edestrans we use the same method we descrbe n secton IV-C. To estmate the future oston of the edestran after T seconds, we select the sequence of sub-goals that has the hghest robablty accordng to Eq. (8). Then as n fgure 8 we comute the ont on the lnes that connect the maxmum robablty sequence of sub-goals located at dstance v T from the start ont. In fgures,, and we show the oston of the nodes n, resectvely, the roosed method, the toology method and the sto ont method. From a qualtatve analyss, we can see that the sub-goal method dects that edestrans manly ust ass through the envronment ( ) but some of them devate from the man corrdor to go to shos (8,3,8), exhbts (6, ) or gates (4,5). The toology method extracts nodes on crossng onts between corrdors (8,,4), but other qualtatvely meanngful onts such as gates or entres of shos are not extracted. The sto ont method has the ooste characterstc: t extracts onts close to gates and shos (,), but ths method cannot extract very well onts on the corrdors. We also notce that the toology method seems sometmes redundant n the descrton of the envronment, resentng often onts very close between them, somethng that does not occur wth the roosed method. Fgure 3 shows an examle that llustrates ths dfference. The area shown n the cture connects the long corrdor and the hallway sace, and thus s a lace where eole are exected to change walng drecton. The yellow lnes vsble on the floor are studded avng blocs located on the tycal walng course to hel the vsually mared, and we observe that usually also healthy eole wal nearby these lnes. Thus, the sub-goal ont s exactly the ont n whch we exect eole to change ther course. When a erson comes from the corrdor (rght sde n the cture), she tycally changes her walng drecton around the ont towards the drecton of sub-goal 9. Also eole comng from the hallway (left sde n the cture) a) Modeled global behavor wth the roosed method. b) Intal oston of sub-goals that are laced at local eas of eq.(6). Fgure. (a) Modeled global behavor n the shong mall A. Flled crcles reresent sub-goal ostons. The sum of transton robabltes between two sub-goals s reresented as the color and the wdth of the ln. Only lns wth transton robablty >. are shown. N=5. (b) Intal ostons of subgoals. Fgure. Grah nodes obtaned wth the toology method. N=33. Fgure. Grah nodes obtaned wth the sto ont method. N=5. : Proosed : Sto ont 9 Fgure 3. Sub-goals and qualtatve onts on the man corrdor drected to the corrdor usually change ther movng drecton at the entrance of the corrdor,.e., around sub-goal. Ths llustrates that sub-goals are extracted n the areas eole tycally go towards, and where they change ther course.

7 On the other hand, the toology and sto ont methods could not extract such onts. The former method rovdes the connecton onts of each corrdor n an envronment, but not those that descrbe mnor devatons n roxmty of shos and the le. Smlarly, the sto ont method could not extract the onts that reresent walng routes through the corrdor, and mostly extracted only onts close to the shos ( and n Fg. 3). Ths method rovdes secfc onts where eole rest, but not onts eole move towards. Judgng from ths qualtatve comarson, we thn that only the roosed method could extract all the mortant nodes of the envronment n a comact (not redundant) way. We also comare the recson n the redcton of future ostons, whch we beleve to be a reasonable quanttatve estmaton of the caablty of methods to descrbe the moton of edestrans. For ths urose, we measure the rato wth whch the correct oston of the edestran s wthn 5m from the estmated oston after T seconds (gven the dmenson of the envronment, a 5 meters error s comatble wth a correct redcton of the area of the envronment the edestran s located n, 5 meters beng roughly the wdth of the long corrdor). Each test traectory s observed for seconds (u to tme t ), and all methods are used to redct the oston at t T, for dfferent values of T (see fgure 4 for a traectory redcton usng the roosed method ). Fgure 5 shows the redcton ratos for all methods. The roosed method erforms clearly better than the sto ont method, wth a dfference that grows wth T, and slghtly better than the toology method. To comare the erformance statstcally, we used a Ch-square test. The comarson wth the sto ont method revealed sgnfcant dfferences (<.5) n the ratos after 8 seconds. The comarson wth the toology method revealed sgnfcant trends (<.) n the ratos at 6 and seconds. In summary, the roosed method not only could extract the qualtatvely mortant onts of the envronment, but also outerformed the other methods from the ont of vew of the recson to redct future ostons. The fact that the erformance of the toology method was close to that of the roosed one suggests that, n the envronment under nvestgaton and for what regards the maor movement along the corrdor, the sub-goals eole are headng to are close to the toologcally sgnfcant onts of the envronment. C. Comarson wth other redcton methods We comare our method also wth two estmaton methods that do not rovde a contnuous grah based descrton of the envronment: a velocty-based lnear extraolaton, and a attern method usng dscrete transton. The velocty based method extraolates the oston at tme t ' based on v, the observed velocty vector at tme t, as x x v ( t' t) (8) t' t Vdeo of the estmaton results can be seen from htt:// * + + Fgure 6. Rato wth whch the estmated oston s wthn 5m from the correct oston (roosed v.s. transton, attern and lnear methods). The transton method uses a robablstc state transton aroach (attern method). The envronment s dvded usng a dscrete grd whose cells are denoted as { g }. A large amount of traectores s observed and dscretzed as a successon of grd states { g } n order to comute the robablty of the n - gram transton ( g g,,..., ) g g (9) n Gven the resent oston x t (located n grd g ), the current,,..., seed v and the dscretzed traectory g g g n, the grd locaton g that maxmze eq. (9) s comuted, and the new oston s gven by T [sec] Proosed Toology Sto ont Fgure 5. Rato wth whch the estmated oston s wthn 5m from the correct oston (roosed v.s. toology and sto ont methods) a) after 8 seconds b) after 3 seconds Fgure 4. Poston and traectory of an observed edestran (blue star and sold lne) and estmaton based on the roosed method (red star and dashed lne). The edestran entered from the rght sde (blue emty crcle), and estmaton s based on observaton of seconds' traectory (u to flled blue crcle). Red crcles reresent hyotheses of estmated ostons and color ntensty reresents robablty (deeer red corresonds to hgh robablty). The hyothess wth hghest robablty s the estmated oston (red star) T [sec] Proosed Pattern Velocty-based

8 xt xt v e () where e s the unt vector drected from x t to the center of grd g, and =5 msec. We name the method attern and use n 6, the same value used for the n -grams n the roosed method. We tred dfferent values for the sze N of the grd and show the results wth the best erformance. To comare the recson n the redcton of future ostons between the roosed method and other methods, we use the same metrc of secton V-B. Fgure 6 shows the ratos for all methods. For comarson we used agan a Ch-square test. The comarson wth the velocty-based lnear extraolaton method revealed sgnfcant dfferences n the ratos after 8 seconds The comarson wth the attern method revealed sgnfcant dfferences n the ratos after 8 seconds (for T =3 sec we have 43% correct redctons wth the roosed method, 9% wth the lnear method and 35% usng the attern method). Thus, the results showed that the roosed method outerforms other methods from the ont of vew of the recson to redct future ostons. VI. CONCLUSIONS In ths aer, we roosed a sub-goal based edestran behavor model and an algorthm that estmates sub-goals n an envronment based on observed traectores of edestrans. The concet of sub-goals as local destnatons of edestran s nown n cogntve scence, but t has not been aled to model edestran behavor n robotcs systems and algorthms have not been develoed to estmate ostons of sub-goals from observed traectores. We roosed an algorthm to estmate the ostons of sub-goals that best descrbe observed edestran traectores. Based on the estmated set of sub-goals, we constructed a edestran model to descrbe the moton of the edestran towards her destnaton. Both the estmated ostons of sub-goals and the edestran movement model are derved from observed real traectores n a shong mall. We used ths model to redct the future oston of edestrans n a large envronment, and comared t to other redcton methods. 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