ageneralizeddynamicpotentialenergymodelformultiagent … · 2019. 12. 3. ·...

Research ArticleA Generalized Dynamic Potential Energy Model for MultiagentPath Planning

Liu He12 Haoning Xi 3 Tangyi Guo 12 and Kun Tang 12

1Department of Automation Nanjing University of Science and Technology Jiangsu 210094 China2MIIT Key Lab of Traffic Information Fusion amp System Control Nanjing China3Research Center for Integrated Transport Innovation (RCITI) School of Civil and Environmental EngineeringUniversity of New South Wales Sydney NSW 2052 Australia

Correspondence should be addressed to Haoning Xi haoningxiunsweduau

Received 3 December 2019 Revised 24 April 2020 Accepted 15 June 2020 Published 24 July 2020

Academic Editor Lu Gao

Copyright copy 2020 Liu He et al is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Path planning for the multiagent which is generally based on the artificial potential energy field reflects the decision-makingprocess of pedestrian walking and has great importance on the field multiagent system In this paper after setting the spatial-temporal simulation environment with large cells and small time segments based on the disaggregation decision theory of themultiagent we establish a generalized dynamic potential energy model (DPEM) for the multiagent through four steps (1)construct the space energy field with the improved Dijkstra algorithm and obtain the fitting functions to reflect the relationshipbetween speed decline rate and space occupancy of the agent through empirical cross experiments (2) Construct the delaypotential energy field based on the judgement and psychological changes of the multiagent in the situations where the otherpedestrians have occupied the bottleneck cell (3) Construct the waiting potential energy field based on the characteristics of themultiagent such as dissipation and enhancement (4) Obtain the generalized dynamic potential energy field by superposing thespace potential energy field delay potential energy field and waiting potential energy field all together Moreover a case study isconducted to verify the feasibility and effectiveness of the dynamic potential energy model e results also indicate that eachagentrsquos path planning decision such as forward waiting and detour in the multiagent system is related to their individualcharacters and environmental factors Overall this study could help improve the efficiency of pedestrian traffic optimize thewalking space and improve the performance of pedestrians in the multiagent system

1 Introduction

Multiagent system used in the simulation reflects thepsychological and physical properties of pedestriansWalking is a hot topic in traffic research in recent years[1] Walking is a sustainable mode in transportation withlittle space no resource consumption and no environ-mental pollution [2] A growing number of recent studieshave focused on improving the sustainability of trans-portation systems by routinely converting motorizedtravel to walking [3] With the general change of theconcept from vehicle-oriented to people-oriented trafficrelated studies on pedestrian traffic are gradually in-creasing In order to identify effective strategies for im-proving the efficiency of pedestrians in a multiagent

system planners need to identify how the current per-sonal characters affect walking in transportation Litera-ture studies [1ndash4] conducted quantitative analysis tooptimize the walking space Literature studies [5ndash7]analysed the importance and effectiveness of walking intransportation through empirical case studies and systemsimulations Eady and Burtt investigated the role ofwalking and access to suburban shopping centres willaffect people traveling around Melbourne [5] Bangchenget al considered the stability of the walking system and therationality of its structural design in the study of dynamicsfor a humanoid robot [6] Yamazaki et al proposed anevacuating agent walking model and the estimated dis-tribution of the evacuation based on the real evacuationdata [7]

HindawiJournal of Advanced TransportationVolume 2020 Article ID 1360491 14 pageshttpsdoiorg10115520201360491

Existing quantitative research on pedestrian trafficmainly focuses on system simulation which can be dividedinto discrete and continuous models Cellular automaton(CA) [8] model is a type of discrete model which dividespedestrian walking areas into cells Social force model is atype of continuous model which defines pedestrians in acontinuous space and describes the pedestriansrsquo behavioursthrough various forces Both models are easy to understandbut cannot reflect the difference among individual charac-teristics of pedestrians Dynamic pedestrian planning is thecore of pedestrian simulation and the main method of pathplanning is to establish a potential energy field [9]e robotmoves in the gradient direction in which the potentialenergy drops fastest to avoid obstacles Deepak Subramaniand Lermusiaux combined the decision theory with theessential stochastic time-optimal path planning to establishthe uncertain strong and dynamic risk-optimal pathplanning scheme based on partial differential equations [10]Antonio Sedentildeo-noda and Colebrook extended the Dijkstraalgorithm to the two-objective shortest path problem [11]Rui Song et al established a smooth A lowast model for pathplanning of unmanned vehicles (UVs) [12] Prahlad et alcombined the genetic algorithm with the global searchfunction and the artificial potential energy field to constructthe EAPF (evolutionary artificial potential field) algorithmthis method is affected by the population number and al-gebra of the genetic algorithm e operation efficiency islow [13] Ulises Orozco-Rosas et al constructed a membraneevolution potential energy field based on a multiprocessorwhich improves the computational efficiency of robot pathplanning [14] Deepak N Subramani et al addressed theoptimal path planning problem in a stochastic dynamicenvironment by combinatorial decision-making and timeoptimization theory [10] ant colony algorithm [15] simu-lated annealing method [16] firefly algorithm [17] andQ-learning [18] e studies mentioned above mainly focuson the optimal path planning in various dynamic scenesunder certain perceptual conditions from the view of systemcontrol and they are not suitable for pedestrian simulationwith physiological psychological and social uncertainties

Pieces of research on pedestrian path planning in themultiagent system belong to the artificial intelligence fieldMultiagent system is composed of multiple interacting el-ements known as agents ey can decide how to satisfytheir objectives and interact with other agents by engaging inanalogues of the kind of social activity cooperation coor-dination and negotiation [19] Since Kinny et al introducedthe multiagent into the air traffic control system [20] themultiagent has been a research focus in the field of trans-portation Compared with vehicles pedestrians have thecharacteristics of blindness (blindness means when thecrowd is highly concentrated especially when the situation isunclear the pedestrian tends to follow the crowd blindly dueto the herd mentality) purpose (purpose means when pe-destrians have a clear destination they will care about fa-cilities guidance systems etc so as to arrive as soon aspossible otherwise it is easier to be attracted by the sur-rounding environmental factors) randomness common-ality and so on Hence it is very important to highlight the

process of independent decision-making and group game inpath planning in a multiagent systemese research studiesproposed several prototype techniques for agent systems intransportation management including methodologies foragent-oriented analysis and design formal specification andverification methods for agent systems and techniques forimplementing agent specifications [21ndash23]

Existing studies only analyze the speed characteristics ofpedestrian flow in the multiagent system [24ndash27] or studythe large-scale macrocollection and distribution capacity ofpedestrians [28ndash30] Few research studies focus on themechanism of path planning for the multiagent at mesoleveland consider different factors such as gender and luggageMoreover the deadlock problem always arises in traditionalcellular automata simulations In order to fill in these re-search gaps this paper constructed the generalized dynamicpotential energy field with large cells and small time seg-ments to improve the discrete artificial potential energy field

To the best of the authorrsquos knowledge few studies es-tablish a dynamic simulation model at the mesolevel toreflect the personal characteristics of each individual in amultiagent system Hence based on the disaggregationcharacteristics of the multiagent system we establish ageneralized dynamic potential energy model (DPEM) effi-ciently and accurately at mesolevel to strengthen the dis-aggregation characteristic of the multiagent system andreflect the characteristics such as gender and carry-onluggage by considering the characteristics of each individualas key parameters e results of the case study validate theeffectiveness of the proposed dynamic potential energymodel (DPEM) and indicate that each agentrsquos path planningdecision such as forward waiting and detour in the mul-tiagent system is related to their characters and environ-mental factors

e rest of the paper is organized as follows Section 2presents the procedure of establishing a generalized dynamicpenitential energy model (DPEM) through 4 steps Section 3conducts a case study to illustrate the feasibility and effec-tiveness of the proposed DPEM Section 4 concludes with asummary of main findings in this research and suggestionsfor future study

2 Methodology

21 Notations and Preliminaries Table 1 summarizes theparameters variables and abbreviations used in this paperto describe the mathematical models

A pedestrian in the simulation system can be regarded asa multiagent HCM2010 defines a pedestrian as an ellipse of046mtimes 061m with a contact area of 028 m2 and anoncontact area of 066 m2 In Figure 1 we measured thespace occupancy of pedestrians of different genders in fivestates of luggage none carry a bag carry a knapsacksuitcase (stationary) and suitcase (moving) To simplify themodel the pedestrians defined in this paper do not includethe elderly and children

In CA models the length of a cell is 01m to 05mWhether the cell is occupied by a pedestrian or obstacles isdenoted as a 0 1 variable In this paper the length of the

2 Journal of Advanced Transportation

cell is set as 08m which is larger than the normal celldefined in CA models and a cell can accommodate multiplepedestrians or obstacles according to its space occupancye pedestrian space of pedestrians is shown in Table 2

In the discrete simulation models an agent can onlyreach the cells in his neighbourhood this paper chooses theMoore neighbourhood and assumes that pedestrians canonly reach one of the eight adjacent directional cells

According to the relevant literature the pedestrianwalking speed is between 04ms and 14ms [31] So ittook 06 ssim2 s for a cell to reach its adjacent vertical cell and08 ssim28 s to the oblique adjacent cell erefore thesimulation time step of the model is set as the maximumcommon divisor 02 s e actual walking time between thecell is commonly determined by the speed of the pedestrianand the number of rows in the cell In this paper we focuson a new type of cell whose length is set as 08m and timestep is set as 02 s and is vividly defined as the ldquopancakecellrdquo

22 Problem Description Walking is a dynamic process ofpedestrian path-planning decisions and displacementsIdeally pedestrians determine their forward direction based

on the shortest distance or time while avoiding other pe-destrians and obstacles Figure 2(a) shows an agentrsquos walkingarea in which the yellow part is an isolation bar ere is agap (11 13) on the west side of the isolation bar which is thebottleneck of the pedestrian flow [32] An agent travels fromthe (2 2) to the blue cell (24 8) During walking gap (10 13)is temporarily occupied by other agents According to thepath planning principle the agent falls into the localminimum point of the artificial potential field At this pointthe agent faces three options

(a) Delay if the delay time is much lower than thedetour time the agent will wait and increase thedelay time in the obstacle cell

(b) Detour if the delay time is larger than the detourtime the agent will change other routes

(c) Waiting an agent will wait in place for a certainperiod before reselection

erefore the dynamic potential energy model (DPEM)for the local minimum problem should reflect three char-acteristics (1) an agent will make choices among delaydetour and waiting (2) an agent will jump out once he fallsinto the local minimum and (3) an agent will not return

Table 1 Parameters variables and abbreviations

q e ID of an agent (i j) e location of the cellk Time step of simulation dir e direction of the agent walking ahead

tk Time of kth time step τij(ρq μ)e velocity decline of agent qwith space occupation ρ walking through the cell with

μρ Space occupation by the agent G(dir) e distance to the adjacent cell by direction dirμ Space occupied by the obstacle αq e waiting potential energy dissipation coefficient of agent qΔμ Change rate of μ during a time step βq e waiting potential energy enhancement coefficient of agent q

vq e expected speed of agent q Vij(q) Space potential energy of agent q in cell (i j)SO Stationary obstacle Tij(q tk) Delay potential energy of agent q in cell (i j) at tk

MG Male group Wij(q tk) Waiting potential energy of agent q in cell (i j) at tk

FG Female group Uij(q tk) Generalized dynamic potential energy of agent q in cell (i j) at tk

Figure 1 Pedestrians of different genders in five states of luggage

Journal of Advanced Transportation 3

back after jumping out of the local minimum cell e effectof delay and waiting cost should be considered besides thestatic space in the potential energy field

23 Model Formulation In order to reflect the actual be-haviours of pedestrians such as delay detour and waitingwe obtain the generalized dynamic potential energy bysuperimposing the delay potential energy waiting potentialenergy and static space potential energy all together enwe investigate the functional relationship between individualattributes and potential energy of the multiagent to reflectpedestriansrsquo different decisions for path planning

231 Space Potential Energy In the space potential energyfield V the spatial potential energy is determined by thedistance from the current position to the destination Sincethe space potential energy is only related to the Euclideandistance and is static it is called the static space potentialenergy

Consider the path planning problem of the multiagent tomultidestinations if there are m agent origins from differentcellsOi i isin m to the destination cell Dj j isin n the traditionalpath search algorithm will calculate mtimes n times in each timestep In order to improve the computational efficiency we canstart from the destination set D to search the cells in allpossible directions and record the distance to any cell

Table 2 Space occupancy of different pedestrians

Gender Luggage Width (m) Distance (m) Area (m2) Space occupancy (ρ) ()

Male

None 055 04 017 27Carry a bag 06 045 021 33

Carry a knapsack 055 05 022 34Suitcase (stationary) 08 05 031 48Suitcase (moving) 08 095 060 94

Female

None 045 03 011 17Carry a bag 055 04 017 27


(a) (b)

(c)

Figure 2 Static artificial potential energy field and walking trajectory


In each step when pedestrians update the current cellcoordinates at least one cell that is nearer to the desti-nation can be found in the neighbourhood Pedestrianswill choose the nearest one as the forward direction andfinally reach the nearest cell in the destination set Since allof the points in this region are search targets the numberof searches is fixed and the Dijkstra algorithm can beimproved directly

Here is the procedure of the improved Dijkstraalgorithm

(i) Step 1 set all cells which meet μgt 1 minus ρ to be minus1 andthe others have no potential energy

(ii) Step 2 traverse all the cells to find the cell b sat-isfying the following conditions

Cell b has no potential energy there is at least onecell with positive potential energy in the neigh-bourhood of the cell e two cells are connectablee cell is assigned a temporary val-ue 1113954Vb Va + Gab where Va is the potential energyof the adjacent cell and Gab is the distance betweentwo adjacent cells whose value is 1 or 1414 If thereis more than one cell in the neighbourhood whichsatisfies the above conditions choose a smallervalue namely 1113954Vb min(Vai

+ Gaib) i 1 2

8 then all the satisfied cells will create a set B

(iii) Step 3 replace the potential energy value with thesmallest temporary value in B Vb 1113954Vb

(iv) Step 4 if all the cells have potential energy endotherwise return to Step 2

e simulation process of the improved Dijkstra algo-rithm is shown in Figure 3

Here we consider a case where an agent will walk fromthe upper left corner to the lower right corner Figure 2(b)displays the distribution of the static potential energy fieldwhere blue colour represents the lower potential energyvalue and red colour represents the higher value Figure 2(c)displays the potential energy and walking trajectory of anagent

232 Delay Potential Energy

(1) Obstacle Delay Function e open space cell has noconstraints on an agentrsquos walking path choice an agent canenter from any direction in the open space cell and walkthrough with the desired speed and thus the initial spaceoccupancy of an open space cell is zero Moreover theobstacles such as walls foundation columns isolation barsand other pedestrians have impacts on an agentrsquos pathchoice e agent will initiatively stay away from the wallsbypass the columns and facilities and keep their distancefrom each other From the perspective of the space occu-pation of the cell the walking environment can be dividedinto four categories

In Table 3 ρ and μ represent the space occupancy of anagent and obstacle respectively e cell with no obstaclesdoes not occupy space and has no effects on path planning ofthe agent e complete obstacle cell is repulsive to the agentfrom any direction A channelized obstacle such as anisolation bar is considered to be at the edge of a specifieddirection of the cell to prevent an agent from walking towardin the specified direction but there are no constraints onpedestrians in other directions

e delay potential energy reflects the time consumptionof pedestrians while passing through the cell In this paper acontinuous variable μisin [0 1] is used to indicate whether thecell is occupied or not instead of 0-1 variables in the tra-ditional methods When an agent with a body size of ρ walksthrough a cell occupying the space of μ there will be delaysdue to lateral avoidance Delays are affected by the followingtwo factors (1) the ldquobody typerdquo of an agent is determined bythe gender and the number of carry-on luggage (2) the spaceoccupation of the cell

en a cross-observation experiment is carried out toexplore the regulation of speed decline rate 40 studentparticipants from the authorrsquos college were evenly dividedinto two groups by gender Place cone buckets of differentsizes or stand on other participant groups to change theoccupancy of the cell every time Each group of partic-ipants is required to carry different luggage through thecell in turn Record the time consumed by each partic-ipant from entering to leaving the cell Divide the timepassed by empty hands by the time passed by this roundto get the speed decline rate If the participant cannot passthe cell it is set as 0 Table 4 shows the results of theexperiment

rough increasing the space occupation of the obstaclesfrom 0 to 100 the functional relationship between speeddecline rate and space occupation for both males and fe-males is fitted and shown in Figure 4

e experimental results (Figure 4) show that thespeed decline rate decreases with the obstacle occupancy(μ) for the agent with different occupancy (ρ) Ifμ + ρ⟶ 1 the rate of speed decline rate will dramaticallyincrease Different types of obstacles have different im-pacts on the speed of the agent among which the sta-tionary obstacle has the least impact on obstaclesrough the experimental observation a male agent has ahigher spatial tolerance to obstacles and can adjust theirposture and luggage position more flexibly to get throughobstacles quickly As a result although the value of ρ forthe male agent is larger than that for the female agent thespeed of a male agent is less affected by obstacles com-pared with that of the female agent Moreover since thespace of the agent with a suitcase is more flexible it ispossible for the agent to walk through the cell whenμ + ρgt 1

Figure 5 displays the curve of τ for male and femaleagents and the fitting formulas of the velocity decline rateτ(ρ μ) for the male agent are given as


Initialize time = 1

Reach the destination

Dir = 1Remaining time = 0

Yes

Remaining time of the grid gt 0

No

Remaining time ndash 1

No

Select the lowest potential energy andpassable grid in the neighbourhood

Yes

Update direction and reset the remaining time

Time + 1

Simulation time is up

Simulation isover

Yes

Record pedestrianrsquos walking

No

Create static potential energy field V

No

No

Update position

ρ + μ of target gridincreases

Figure 3 Flowchart of the improved Dijkstra algorithm

Table 3 Obstacle classification

Type Space occupation μ Impact on the cell ExampleNo obstacle μ 0 None Open spaceComplete obstacle μge 1 minus ρ Prevents entry from any direction WallsCanalized obstacle μ 0 Prevents entry from a particular direction Isolation barPartial obstacle 0lt μlt 1 minus ρ Slows down the walking speed Dustbin


τ(ρ μ)

1 μ 0

0 ρ + μge 1

minus1452(ρ + μ)2 + 05893(ρ + μ) + 1 type SO ρ + μlt 1

minus1297(ρ + μ)2 + 0343(ρ + μ) + 1 type MG ρ + μlt 1

minus1616(ρ + μ)2 + 04191(ρ + μ) + 1 type FG ρ + μlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

e fitting formulas of the velocity decline rate τ(ρ μ)

for the female agent are given as

τ(ρ μ)

1 μ 0

0 ρ + μge 1


minus0795 ln(ρ + μ) minus 00748 type MG ρ + μlt 1

minus06275(ρ + μ)2 minus 03902(ρ + μ) + 1 type FG ρ + μlt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

As shown in Figure 5 all five curves had R2 fittingvalues greater than 09 except for the curve representingwomen crossing the male group (MG) and this may bebecause women are more repelled by the opposite gendererefore the time that the agent reaches any adjacentcell can be expressed as

Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)

where G(dir) represents the distance to the adjacent cell andis related to the direction of the agent and vq denotes theexpected speed of agent q

(2) Delay Potential Energy Function e static potentialenergy value is determined by the spatial distance betweenthe cell and the destination e pedestrian chooses the cellwith the lower potential energy value which is closer to thedestination Assuming that the pedestrians will walk at the

Table 4 Relationship between speed decline rate and space occupation

Sex Luggage ρ () Obstacle typeObstacle occupation μ

0 10 20 30 40 50 60 70 80 90 100

Male

None 27

Stationary obstacle (SO)

100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0

Knapsack 34 100 95 91 80 71 57 33 7 1 0 0Suitcase 94 100 61 24 11 3 2 0 0 0 0 0

Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0



expected speed the shortest path is considered both theshortest distance and time to unify the time and space

After considering the influence of obstacles the actualspeed of the agent is gradually lower than the expected speedand the relationship between walking distance and time isnonlinear (Figure 6) In this case shortening the walkingdistance will increase the time the agent needs to walk formore distance to save the time and thus the multipathproblem will arise erefore the microscopic path-planningproblem can be transformed into amultiobjective optimizationproblem considering the influence of partial obstacles ereare two methods to solve the multiobjective problem (1) totransform the multiobjective problem into a single objectiveproblem by weighting (2) to solve the problem based on thePareto optimal solutions such as data envelopment analysis(DEA) and heuristic algorithm Since the two objectives of time

and space distance can be converted through velocity the firstmethod is adopted in this paper

e agent often makes different choices on the time andspace distance From experience the agent in a hurry willchoose the most time-saving path while others will choose arelatively short path We explain this phenomenon as theldquodegree of lazinessrdquo and the generalized potential energy of theagent is that spatial distance and time are weighted by ldquolazinesscoefficientrdquoeweighting factor of the distance is larger for theldquolazierrdquo agent and the weighting factor of the time is larger forthe diligent agent so the spatial potential energy can betransformed into the generalized potential energy In this waythe choice of path for the agent is more objective and diverse

en we explain the ldquolaziness coefficientrdquo from the viewof opportunity cost where the difference lies in the value oftime (VOT) of the agent Moreover the opportunity cost of

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)

Knapsack (ρ = 034)Suitcase (ρ = 094)

The male agent walks through stationary obstacle (SO)

(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)


The female agent walks through stationary obstacle (SO)

(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)


The male agent walks through male group (MG)

(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)


The female agent walks through male group (MG)

(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)


The male agent walks through female group (FG)

(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)


The female agent walks through female group (FG)

(f )

Figure 4 Experimental curve of speed drop and space occupation


time is reflected by the maximum distance that the agent canwalk during the delayed time period

us waiting potential energyTij(q) can be expressed as

Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)

(4)where vq represents the expected speed of agent q andτij(ρq μ) represents the coefficient of the velocity declinerate for the partial obstacle in the cell (i j)

233 Waiting Potential Energy Since the agent will losepatience and choose other paths after waiting if other pe-destrians have occupied the target cell the waiting potential

energy is introduced in this paper e waiting potentialenergy is affected by the characteristics of the agent andchanges with time and has two opposite characteristics ofdissipation and enhancement

(1) Dissipation If the pedestrian has left the cell (i j) thewaiting potential energy W will dissipate over time At firstthe agent still remembers that he has passed through the cell(i j) and thus the waiting potential energy can keep pe-destrians from returning When the waiting potential energydissipates to 0 the agent will forget whether he has passedthrough the cell

When agent q leaves the cell (i j) at t0 the waitingpotential energy of the cell (i j) can be expressed asWij(q tk) after k simulation time steps

120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG

Fitting curve of male agent

(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG

y = ndash0795ln(x) ndash 00748R2 = 08318

FGy = ndash06275x2 ndash 03902x + 1

R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG

Fitting curve of female agent

(b)

Figure 5 Fitting curve of τ (a)e relationship between speed decline rate (τ) and (b) space occupation (μ + ρ) for male and female agentsrespectively


Wij q tk( 1113857 αqWij q tkminus1( 1113857 α2qWij q tkminus2( 1113857

αkqW q t0( 1113857 0lt αq lt 1

(5)

where W(q t0) is the basic waiting potential energy and αq isthe waiting potential energy dissipation coefficient of agentq e smaller αq is the faster the waiting potential energywill dissipate

e waiting potential energy W(q tk) is mainly affectedby three variables basic waiting potential energy W(q t0)dissipation coefficient αq and enhancement coefficient βq

If the agent waits for a time step he will give up themovement during this time step at the expected speed andthis is also defined as the opportunity cost

W q t0( 1113857 02 middot vq tk( 1113857 (6)

e dissipation coefficient αq reflects the perception ofthe agent If αq⟶ 1 it will dissipate slowly and it is

impossible for the agent to return If αq⟶ 0 the agent willreturn to the same local minimum cell and fall into a deadcycle From this point of view αq should be as large aspossible However the local minimum of the cell maydisappear after a short period since the agent blocking at thedoor will leave it will take a long time for the agent to returnerefore the dissipation coefficient also reflects how far theagent is willing to search after falling into a local minimumand successfully jumping out According to observationexperiments the interval between pedestrian path-planningdecisions is about 1 second and it is related to the genderage and luggage of the agent To simplify the model letαq 09 after 1 second (5 time steps) the waiting potentialenergy can be reduced to the original 095 059 timesnamely it will dissipate at a rate of 60 per second

(2) Enhancement If the agent stays in the cell (i j) thewaiting potential energy W will increase over time and

Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0

Figure 6 Relationship between time and walking distance

Table 5 Simulation process of multiagent path planning

Step Description1 Model initialization Set the generation rate of the agent as αo and the number of agents as k2 Obtain the static spatial potential energy field V for destination set through the improved Dijkstra algorithm3 Generalize the delayed potential energy field T for all agents set the potential energy field W 0 and simulation time t 14 Wait for the dissipation of potential energy W5 Set the agent number as k 16 Select the kth agent and jump to Step 13 if the agent has not departed or left the simulation area Otherwise move to Step 7

7 If the agent is not waiting and the remaining time of the cell is greater than 0 then the remaining time will be minus1 and jump to Step 13Otherwise move on to Step 8

8 If the agent reaches the destination set the agent waiting subtract the space occupation of the agent from the space occupation of thecell if the waiting potential energy field W becomes 0 then jump to Step 13 otherwise move on to Step 9

9 If the current time equals the initial entry time of the agent strengthen the space occupation in the origin cell Otherwise if thedirection of the agent is 1 the waiting potential energy will be strengthened

10 Strengthen the basic waiting potential energy and update 3 types of potential energy in all directions to obtain the generalizedpotential energy U by superposition

11 Determine the target cell according to the cell probability selection model12 If the cell is a target cell the agent will choose to stay and set the agent waiting Otherwise update the direction and remaining time13 If k K move on to the next step Otherwise k k + 1 and return to Step 614 If t tmax end Otherwise t t + 1 and return to Step 4


the agent will wait before falling into a local minimume longer the waiting time is the greater the waitingpotential energy of the cell will be When the super-imposed potential energy of the cell is higher than that ofother cells in the neighbourhood the agent will

successfully jump out of the local minimum On thecontrary after jumping out of the local minimum thewaiting potential energy of the cell is large and dissipatesslowly so pedestrians will not return in a short timeWhen pedestrian q enters and stays in the cell (i j) at t0

Initialize t = 1

Environmental property

Reach thedestination

Wait and the remaining time of the

cell is 0 update μ

Yes

Remaining time gt 0wait or notRemaining time ndash 1

Yes

Yes


Yes

Time is up

Simulation is over

Yes

Record the walking trajectories of the agents

Generation rate of the agents

Select the kth agent

In the simulation area

Yes

Leave the cellYes

Select the next cell

Update all potential energy fields and superpose them to obtain the generalized potential energy field U

Traverse all the agents

Create potential energy field V delayed potential energy field T and set the waiting potential energy field W as 0

Initialize the sequence k = 1

Waiting potential energy field W will dissipate

Strengthen potential energy field W

Wait

No

Wait

No

No

Strengthen waiting potential energy

k + 1No

t + 1No

No

First entry

No

Update μYes

Create property table

No

Figure 7 Flowchart of multiagent path planning


after k simulation time steps the waiting potential energyof the cell (i j) is expressed as

Wij q tk( 1113857 βqWij q tkminus1( 1113857 + Wij q t0( 1113857

1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

where βq is the waiting potential energy enhancement co-efficient of agent q e larger value of βq represents that theagent is reluctant to wait When βq 1 the waiting potentialenergy increases linearly with time when βq gt 1 the longerthe pedestrians wait the faster the waiting potential energyincreases When the total potential energy of the super-imposed cell is greater than that of the neighbourhood cellpedestrians successfully jump out of the local minimumis can also reflect the process that pedestrians graduallylose patience and look for other paths after waiting for aperiod

e enhancement coefficient βq reflects the patience ofthe agent e lower value of the coefficient means that theagent is more patient e decisive factor of βq is the agentrsquosjudgement on velocity decline rate of the occupation in thebottleneck cell which is also related to the gender age andtravel purpose of the agent In order to simplify this model itis assumed that βq is only related to the variation of oc-cupation μ in the next time step

Impatient agent judges that the cell occupation of obstaclesis constant and will leave at the next time step after jumping intothe local minimum cell and thus whenΔμ μ(tk+1) minus μ(tk) 0 βq will reach themaximumβq +infin e patient agent thinks that the occu-pancy of the obstacle cell will be reduced to zero and thus whenΔμ μ(tk+1) minus μ(tk) μ βq reaches the minimum valueβq 1 erefore the logarithmic function can be established

βq lnμΔμ

+ 1 μgeΔμ (8)

234 Generalized Dynamic Potential Energy e general-ized dynamic potential energy Uij(q tk) can be obtained bysuperposing the space potential energyVij(q) delay po-tential energy Tij(q tk) and waiting potential energyWij(q tk) Moreover three types of potential energy areunified by distance

Uij q tk( 1113857 Vij(q) + Tij q tk( 1113857 + Wij q tk( 1113857 (9)

e simulation process of the multiagent is shown inTable 5 and Figure 7

3 Case Study

After superposing the delay potential energy field T andwaiting potential energy fieldW consider the path planningproblem of a male agent without luggage (ρ 027） underdifferent space occupation of obstacles ( μ) e resultsobtained from the DPEM state that the agent will makedifferent choices while facing the local minimum cell withdifferent values of μ

If μ 02 the agent will choose to go through the cell ifμ 09 the agent will choose to detour affected by conflictsbetween the potential energy and the inertia potential energyof static obstacles

In Figure 8(a) the agent falls into the local minimum cell(10 13) at the 75th time step After delaying 4 time steps theagent will jump out at the 83rd time step and delay passingthrough the cell (11 13) where the obstacle is located andreach the destination at the 158th time step

In Figure 8(b) the agent falls into the local minimum cell(10 13) at the 75th time step After waiting for 5 time stepsthe agent jumps out at the 80th time step and choose todetour reaching the destination at the 214th time step

4 Conclusions

is paper first proposes a dynamic potential energy model(DPEM) under the spatial-temporal simulation environ-ment with large cells and small time segments where thedeadlock problem arisen in traditional cellular automatasimulations can be avoided Secondly this paper concludes

(a) (b)

Figure 8 Testing the local minimum detour (a) and (b) walking trajectories of the agent where μ 02 and μ 09 respectively


that the agentrsquos selection such as forward waiting anddetour while facing the obstacles is a dynamic decision-making process Each agent will decide on opportunity costwith the objective of utility maximization and is affected bytheir personal characters and other environmental factorsirdly through the empirical experiments it is showed thatthe sum of the agent and obstaclesrsquo space occupation (ρ + μ)is the main factor affecting the velocity decline rate in themultiagent systeme velocity decline rate is also related tothe obstacle types such as the stationary objective (SO) malegroup (MG) and female group (FG) Moreover it is showedthat the trajectory of each agent in the multiagent system hastwo characteristics dissipation and enforcement e dis-sipation characteristic can guarantee the condition in whicheach agent will not return and the enforcement charac-teristic can guarantee the condition in which each agent willfirst wait and then make detours while facing the obstaclesen the relationship between each agentrsquos patience whilewaiting and the space occupation change rate of obstacles isshowed to be logarithmic Finally we conduct a case study toverify the effectiveness of the proposed DPEM e researchresults of this paper will be helpful for the construction of amesoscopic pedestrian traffic simulation model accuratelyand efficiently and reflect different characters of each agentsuch as the gender and carry-on baggage In the futureresearch we will introduce more parameters and trafficcontrol measures such as travel companion traffic signalspedestriansrsquo selection between stairs and escalators to de-scribe the environmental factors and group characters of themultiagent based on the technology of data mining anddecision-making game theory

Data Availability

e experimental data used to support the findings of thisstudy are included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e Fundamental Research Funds for the Central Univer-sities (30919011290)

References

[1] E Porter S H Hamdar and W Daamen ldquoPedestrian dy-namics at transit stations an integrated pedestrian flowmodeling approachrdquo Transportmetrica A Transport Sciencevol 14 no 5-6 pp 468ndash483 2018

[2] T Saghapour S Moridpour and R ompson ldquoSustainabletransport in neighbourhoods effect of accessibility on walkingand bicyclingrdquo Transportmetrica A Transport Science vol 15no 2 pp 849ndash871 2019

[3] I M Lee and D M Buchner ldquoe importance of walking topublic healthrdquo Medicine amp Science in Sports amp Exercisevol 40 no 7 pp S512ndashS518 2008

[4] R Gerike A de Nazelle R Wittwer and J Parkin ldquoSpecialissue ldquowalking and cycling for better transport health and theenvironmentrdquo Transportation research Part A Policy andpractice vol 123 2019

[5] J Eady and D Burtt Walking and Transport in MelbourneSuburbs Victoria Walks Incorporated Victoria Australia2019

[6] B Zhang C Shao Y Li H Tan and D Jiang ldquoDynamicsimulation analysis of humanoid robot walking system basedon ADAMSrdquo Journal of Shanghai Jiaotong University (Sci-ence) vol 24 no 1 pp 58ndash63 2019

[7] T Yamazaki N Kobayashi Y Owada and G Sato ldquoAgentwalking model construction in urban disaster simulationrdquo inProceedings of the 2018 IEEE 7th Global Conference on Con-sumer Electronics (GCCE) IEEE Nara Japan pp 355-356October 2018

[8] C Dias and R Lovreglio ldquoCalibrating cellular automatonmodels for pedestrians walking through cornersrdquo PhysicsLetters A vol 382 no 19 pp 1255ndash1261 2018

[9] O Khatib Real-Time Obstacle Avoidance For ManipulatorsAnd Mobile Robots pp 396ndash404 Springer New York NYUSA 1986

[10] D N Subramani and P F J Lermusiaux ldquoRisk-optimal pathplanning in stochastic dynamic environmentsrdquo ComputerMethods in Applied Mechanics and Engineering vol 353pp 391ndash415 2019

[11] A Sedentildeo-noda and M Colebrook ldquoA biobjective Dijkstraalgorithmrdquo European Journal of Operational Researchvol 276 no 1 pp 106ndash118 2019

[12] R Song Y Liu and R Bucknall ldquoSmoothed Alowast algorithm forpractical unmanned surface vehicle path planningrdquo AppliedOcean Research vol 83 pp 9ndash20 2019

[13] P Vadakkepat K C Tan and W Ming-Liang ldquoEvolutionaryartificial potential fields and their application in real timerobot path planningrdquo in Proceedings of the 2000 congress onevolutionary computation vol 1 IEEE La Jolla CA USApp 256ndash263 2000 July

[14] U Orozco-Rosas O Montiel and R Sepulveda ldquoMobilerobot path planning using membrane evolutionary artificialpotential fieldrdquo Applied Soft Computing vol 77 pp 236ndash2512019

[15] Z Zhishui ldquoAnt colony algorithm based on path planning formobile agent migrationrdquo Procedia Engineering vol 23pp 1ndash8 2011

[16] H Miao and Y-C Tian ldquoDynamic robot path planning usingan enhanced simulated annealing approachrdquo AppliedMathematics and Computation vol 222 pp 420ndash437 2013

[17] B K Patle A Pandey A Jagadeesh and D R Parhi ldquoPathplanning in uncertain environment by using firefly algo-rithmrdquo Defence Technology vol 14 no 6 pp 691ndash701 2018

[18] E S Low P Ong and K C Cheah ldquoSolving the optimal pathplanning of a mobile robot using improved Q-learningrdquoRobotics and Autonomous Systems vol 115 pp 143ndash161 2019

[19] M Wooldridge An Introduction to Multiagent Systems JohnWiley amp Sons Hoboken NY USA 2009

[20] D Kinny and M Georgeff ldquoModelling and design of multi-agent systemsrdquo in Proceedings of the International Workshopon Agent Geories Architectures and Languages SpringerBerlin Germany pp 1ndash20 1996 August

[21] M Wooldridgey and P Ciancarini ldquoAgent-oriented softwareengineering the state of the artrdquo in Proceedings of the In-ternational Workshop on Agent-Oriented Software Engineer-ing Springer Berlin Heidelberg pp 1ndash28 2000 June


[22] L Okdinawati T M Simatupang and Y Sunitiyoso ldquoMulti-agent reinforcement learning for collaborative transportationmanagement (ctm)rdquo in Proceedings of the Agent-Based Ap-proaches in Economics and Social Complex Systems IX SpringerSingapore pp 123ndash136 2017

[23] A Baykasoglu V Kaplanoglu and C Sahin ldquoRoute prioriti-sation in a multi-agent transportation environment via multi-attribute decision makingrdquo International Journal of DataAnalysis Techniques and Strategies vol 8 no 1 pp 47ndash64 2016

[24] C Liao H Guo K Zhu and J Shang ldquoEnhancing emergencypedestrian safety through flow rate design bayesian-nashequilibrium in multi-agent systemrdquo Computers amp IndustrialEngineering vol 137 p 106058 2019

[25] C Yu G Ren and T Zhang ldquoSimulation of pedestrian flow ata crosswalk by a multi-agent system with a pre-avoid forcesmodelrdquo in Proceedings of the CICTP 2016 pp 2303ndash2313Shanghai China July 2016

[26] J Ren W Xiang Y Xiao R Yang D Manocha and X JinldquoHeter-Sim heterogeneous multi-agent systems simulationby interactive data-driven optimizationrdquo IEEE Transactionson Visualization and Computer Graphics 2019

[27] P Stone and M Veloso ldquoMultiagent systems a survey from amachine learning perspectiverdquo Autonomous Robots vol 8no 3 pp 345ndash383 2000

[28] M Hussein and T Sayed ldquoValidation of an agent-basedmicroscopic pedestrian simulation model in a crowded pe-destrian walking environmentrdquo Transportation Planning andTechnology vol 42 no 1 pp 1ndash22 2019

[29] S Tak S Kim and H Yeo ldquoAgent-based pedestrian celltransmission model for evacuationrdquo Transportmetrica ATransport Science vol 14 no 5-6 pp 484ndash502 2018

[30] K R Rozo J Arellana A Santander-Mercado and M Jubiz-Diaz ldquoModelling building emergency evacuation plansconsidering the dynamic behaviour of pedestrians usingagent-based simulationrdquo Safety Science vol 113 pp 276ndash2842019

[31] C Y Cheung and W H K Lam ldquoPedestrian route choicesbetween escalator and stairway in MTR stationsrdquo Journal ofTransportation Engineering vol 124 no 3 pp 277ndash285 1998

[32] O Handel and A Borrmann ldquoService bottlenecks in pe-destrian dynamicsrdquo Transportmetrica A Transport Sciencevol 14 no 5-6 pp 392ndash405 2018


Existing quantitative research on pedestrian trafficmainly focuses on system simulation which can be dividedinto discrete and continuous models Cellular automaton(CA) [8] model is a type of discrete model which dividespedestrian walking areas into cells Social force model is atype of continuous model which defines pedestrians in acontinuous space and describes the pedestriansrsquo behavioursthrough various forces Both models are easy to understandbut cannot reflect the difference among individual charac-teristics of pedestrians Dynamic pedestrian planning is thecore of pedestrian simulation and the main method of pathplanning is to establish a potential energy field [9]e robotmoves in the gradient direction in which the potentialenergy drops fastest to avoid obstacles Deepak Subramaniand Lermusiaux combined the decision theory with theessential stochastic time-optimal path planning to establishthe uncertain strong and dynamic risk-optimal pathplanning scheme based on partial differential equations [10]Antonio Sedentildeo-noda and Colebrook extended the Dijkstraalgorithm to the two-objective shortest path problem [11]Rui Song et al established a smooth A lowast model for pathplanning of unmanned vehicles (UVs) [12] Prahlad et alcombined the genetic algorithm with the global searchfunction and the artificial potential energy field to constructthe EAPF (evolutionary artificial potential field) algorithmthis method is affected by the population number and al-gebra of the genetic algorithm e operation efficiency islow [13] Ulises Orozco-Rosas et al constructed a membraneevolution potential energy field based on a multiprocessorwhich improves the computational efficiency of robot pathplanning [14] Deepak N Subramani et al addressed theoptimal path planning problem in a stochastic dynamicenvironment by combinatorial decision-making and timeoptimization theory [10] ant colony algorithm [15] simu-lated annealing method [16] firefly algorithm [17] andQ-learning [18] e studies mentioned above mainly focuson the optimal path planning in various dynamic scenesunder certain perceptual conditions from the view of systemcontrol and they are not suitable for pedestrian simulationwith physiological psychological and social uncertainties

Pieces of research on pedestrian path planning in themultiagent system belong to the artificial intelligence fieldMultiagent system is composed of multiple interacting el-ements known as agents ey can decide how to satisfytheir objectives and interact with other agents by engaging inanalogues of the kind of social activity cooperation coor-dination and negotiation [19] Since Kinny et al introducedthe multiagent into the air traffic control system [20] themultiagent has been a research focus in the field of trans-portation Compared with vehicles pedestrians have thecharacteristics of blindness (blindness means when thecrowd is highly concentrated especially when the situation isunclear the pedestrian tends to follow the crowd blindly dueto the herd mentality) purpose (purpose means when pe-destrians have a clear destination they will care about fa-cilities guidance systems etc so as to arrive as soon aspossible otherwise it is easier to be attracted by the sur-rounding environmental factors) randomness common-ality and so on Hence it is very important to highlight the

process of independent decision-making and group game inpath planning in a multiagent systemese research studiesproposed several prototype techniques for agent systems intransportation management including methodologies foragent-oriented analysis and design formal specification andverification methods for agent systems and techniques forimplementing agent specifications [21ndash23]

Existing studies only analyze the speed characteristics ofpedestrian flow in the multiagent system [24ndash27] or studythe large-scale macrocollection and distribution capacity ofpedestrians [28ndash30] Few research studies focus on themechanism of path planning for the multiagent at mesoleveland consider different factors such as gender and luggageMoreover the deadlock problem always arises in traditionalcellular automata simulations In order to fill in these re-search gaps this paper constructed the generalized dynamicpotential energy field with large cells and small time seg-ments to improve the discrete artificial potential energy field

To the best of the authorrsquos knowledge few studies es-tablish a dynamic simulation model at the mesolevel toreflect the personal characteristics of each individual in amultiagent system Hence based on the disaggregationcharacteristics of the multiagent system we establish ageneralized dynamic potential energy model (DPEM) effi-ciently and accurately at mesolevel to strengthen the dis-aggregation characteristic of the multiagent system andreflect the characteristics such as gender and carry-onluggage by considering the characteristics of each individualas key parameters e results of the case study validate theeffectiveness of the proposed dynamic potential energymodel (DPEM) and indicate that each agentrsquos path planningdecision such as forward waiting and detour in the mul-tiagent system is related to their characters and environ-mental factors

e rest of the paper is organized as follows Section 2presents the procedure of establishing a generalized dynamicpenitential energy model (DPEM) through 4 steps Section 3conducts a case study to illustrate the feasibility and effec-tiveness of the proposed DPEM Section 4 concludes with asummary of main findings in this research and suggestionsfor future study

2 Methodology

21 Notations and Preliminaries Table 1 summarizes theparameters variables and abbreviations used in this paperto describe the mathematical models

A pedestrian in the simulation system can be regarded asa multiagent HCM2010 defines a pedestrian as an ellipse of046mtimes 061m with a contact area of 028 m2 and anoncontact area of 066 m2 In Figure 1 we measured thespace occupancy of pedestrians of different genders in fivestates of luggage none carry a bag carry a knapsacksuitcase (stationary) and suitcase (moving) To simplify themodel the pedestrians defined in this paper do not includethe elderly and children

In CA models the length of a cell is 01m to 05mWhether the cell is occupied by a pedestrian or obstacles isdenoted as a 0 1 variable In this paper the length of the


























Male

None 055 04 017 27Carry a bag 06 045 021 33


Female

None 045 03 011 17Carry a bag 055 04 017 27


(a) (b)

(c)








+ Gaib) i 1 2















Initialize time = 1



Yes


No


No


Yes


Time + 1


Simulation isover

Yes


No


No

No

Update position






τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)





0 10 20 30 40 50 60 70 80 90 100

Male

None 27


100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0


Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0








100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References



























































Male

None 055 04 017 27Carry a bag 06 045 021 33


Female

None 045 03 011 17Carry a bag 055 04 017 27


(a) (b)

(c)








+ Gaib) i 1 2















Initialize time = 1



Yes


No


No


Yes


Time + 1


Simulation isover

Yes


No


No

No

Update position






τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)





0 10 20 30 40 50 60 70 80 90 100

Male

None 27


100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0


Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0








100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References








































+ Gaib) i 1 2















Initialize time = 1



Yes


No


No


Yes


Time + 1


Simulation isover

Yes


No


No

No

Update position






τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)





0 10 20 30 40 50 60 70 80 90 100

Male

None 27


100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0


Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0








100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References



































Initialize time = 1



Yes


No


No


Yes


Time + 1


Simulation isover

Yes


No


No

No

Update position






τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)





0 10 20 30 40 50 60 70 80 90 100

Male

None 27


100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0


Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0








100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References



































τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



τ(ρ μ)

1 μ 0

0 ρ + μge 1




⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


Tij(q) G(dir)

vqτij ρq μ1113872 1113873 (3)





0 10 20 30 40 50 60 70 80 90 100

Male

None 27


100 98 95 84 76 61 40 10 1 0 0Bag 33 100 93 90 78 67 52 21 3 1 0 0


Female

None 17 100 96 90 81 72 54 35 12 2 0 0Bag 27 100 92 88 75 62 47 15 2 0 0 0


Male

None 27

Male group (MG)

100 95 88 75 63 49 33 10 3 1 0Bag 33 100 90 80 69 54 38 15 5 2 0 0


Female

None 17 100 90 78 47 21 7 2 0 0 0 0Bag 27 100 85 70 41 18 5 1 0 0 0 0


Male

None 27

Female group (FG)

100 95 87 69 47 30 8 0 0 0 0Bag 33 100 92 82 60 38 15 5 0 0 0 0


Female

None 17 100 94 82 72 54 41 30 12 5 0 0Bag 27 100 88 75 58 41 29 12 2 1 0 0








100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References








































100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 027)Bag (ρ = 033)



(a)

100908070605040μ ()

3020100

120

100

80

τ (

)

60

40

20

0

None (ρ = 017)Bag (ρ = 027)



(b)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(c)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(d)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 027)Bag (ρ = 033)



(e)

120

100

80

τ (

)

60

40

20

0100908070605040

μ ()3020100

None (ρ = 017)Bag (ρ = 027)



(f )





Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References





































Tij(q) G(dir)

τij ρq μ1113872 1113873vq

minusG(dir)

vq

⎛⎝ ⎞⎠vq 1 minus τij ρq μ1113872 1113873

τij ρq μ1113872 1113873G(dir)






120

100

80

60τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOy = ndash1452x2 + 05893x + 1

R2 = 09645MG

y = ndash1297x2 + 0343x + 1R2 = 09928

FGy = ndash1616x2 + 04191x + 1

R2 = 0969

SOMGFG

SOMGFG


(a)

SOy = ndash11947x2 + 01656x + 1

R2 = 09246MG



R2 = 09012

120

100

80

60

τ (

)

40

20

0 20 40 60

ρ + μ ()

80 100 1200

ndash20

SOMGFG

SOMGFG


(b)





(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References





































(5)




W q t0( 1113857 02 middot vq tk( 1113857 (6)




Walking distance

Expected speed

Opportunity cost

Actual speed

Delay

Time0












Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References





































Initialize t = 1




cell is 0 update μ

Yes


Yes

Yes


Yes

Time is up

Simulation is over

Yes





Yes

Leave the cellYes








Wait

No

Wait

No

No


k + 1No

t + 1No

No

First entry

No

Update μYes


No





1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References





































1 minus βk+1q

1 minus βq

W q t0( 1113857 βq gt 1

kW q t0( 1113857 βq 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)




βq lnμΔμ

+ 1 μgeΔμ (8)




3 Case Study





4 Conclusions


(a) (b)




Data Availability




Acknowledgments


References




































Data Availability




Acknowledgments


References



































ageneralizeddynamicpotentialenergymodelformultiagent … · 2019. 12. 3. ·...

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