Jamming phenomena of self-driven particles

Transcription

1 Jamming phenomena of self-driven particles Pedestrian Outflow and Obstacle Walking with Slow Rhythm Daichi Yanagisawa, RCAST, UTokyo

2 Pedestrian Outflow and Obstacle Phys. Rev. E, 76(6), , 2007 Phys. Rev. E, 80(3), , 2009 SICE Journal of Control, Measurement, and System Integration, 3(6), pp , 2010

3 Pedestrian Outflow (POF) 1[m] Time = POF= [sec] [persons/(m sec)] Pedestrian Outflow (POF) is the number of pedestrians, who go through an 1 [m] exit in 1 [sec]. POF greatly affects Total Evacuation Time. Large POF = Small Total Evacuation Time. Motivation 1: How to estimate POF for various cases? Motivation 2: How to increase POF?

4 2 1 E pp Floor Field Cellular Automaton pp 5 Model for Evacuation Time step = 2 5 O O O Hopping probability pp dir = NN exp kk ss SS dir dir =,,,, stay NN: Normalization kk ss : Sensitivity parameter kk ss SS dir : Static floor field (Distance to the exit) Only one pedestrian can enter in one cell.

5 Exit Simulation Does not important Important!

6 Mathematical Formulation for POF 1 1 β β 2 E β ββττ θθ β 1 β k There are nn pedestrians around the Exit cell. kk pedestrians are trying to move to there. No cell structure (lattice). (nn determines the cell structure.) POF: qq nn, θθ = 1 rr nn + 1 nn nnnn mm=1 1 ττ θθ mm 1 nn rr nn = 1 φφ ζζ kk bb kk bb kk = kk=1 nn kk ββkk 1 ββ nn kk

7 Two Important Factors in Evacuation 1. Conflicts through an Narrow Exit Friction Function φφ ζζ kk = 1 1 ζζ kk kkkk 1 ζζ kk 1 nn = 3 2. Turning ζζ 0,1 : Aggressive parameter kk N: Number of pedestrians move to an exit at the same time When pedestrians turn θθ at the exit cell, their walking speed decrease. β βτθ ( ) Turning Function ττ θθ = exp ηη θθ 0,1 ηη 0, : Inertia coefficient θθ ππ, ππ : Turning angle

8 Schematic View of the Experiments cm Exit Width (A) n=1 π (B) n=2 6 0 π 2 (C) n=2 50cm π 2 π 4 0 (D) n=2 (E) n=3 (F) n=3 π 2 0 Evacuees 18 people π 2 π cm 2 or 3 times each (G) n=4 (H) (I)

9 Experiments

10 Theory v.s. Experiment Zipper merging (Almost 1 lane)

11 Quiz Which is the fastest? Exit Normal nn = 44 Shifted Obstacle nn = line nn = 11 Center Obstacle nn = 55 44

12 Experiment Normal POF=2.78 Shifted Obstacle POF=2.92

13 Walking with Slow Rhythm h b r i r o N pedestrians Phys. Rev. E, 85(1), , PED 2012, pp , 2014.

14 Music Tempo v.s. Pace Walking on Music Pace v.s. Velocity Walking Pace [BPM] Music Tempo Pace Walking Velocity [km/h] Velocity Pace Music Tempo [BPM] Walking Pace [BPM] (BPM = Beat per Minutes) Frederik Styns et al, Walking on Music, Human Movement Science, 26, 769, Rhythm can control walking velocity of single pedestrian. What happens in a crowd case?

15 b h r i r o Circuit and Model Parameters Length of the circuit: LL Density ρρ = LL NN Headway Distance h = LL bbbb NN h ρρ = 1 bbbb ρρ N pedestrians Assumptions Overtaking is prohibited. Both characteristics and distribution of pedestrians are homogeneous.

16 Stride Function S and Pace Function P Stride S, Pace P Stride S Pace P ρ c Density persons m Density ρρ [person/m] Low-density regime ρρ 0, ρρ cc SS ρρ = ss PP ρρ = pp ρ j VV ρρ = SS ρρ PP(ρρ) Velocity Stride Pace b=1, s=2, k=1, p=1, a=0.2 ρc = k, ρ j kb + s High-density regime ρρ ρρ cc, ρρ jj SS ρρ = kkk(ρρ) PP ρρ = pp aa h ρρ cc h(ρρ) kk (0,1]: Effect of personal space aa kkkk/ss: Effect of density on pace = 1 b

17 Normal and Rhythmic Walking Normal Walking a>0 Pace decreases in the high-density regime. Rhythmic Walking a=0 Pace does not change in the high-density regime. Stride S, Pace P Stride S, Pace P Stride S Pace P Density persons m ρρ [person/m] Stride S Pace P Density persons m ρρ [person/m]

18 persons sec Flow QQ [persons/sec] Flow Q Theoretical Analysis Convex Downward Density persons m ρρ [person/m] Fast Rhythm Normal Slow Rhythm (p, a) (1.2, 0) (1, 0.5) (0.8, 0) Fast Rhythm increases the pedestrian flow. Slow Rhythm improves the flow in the high-density regime.

19 Single Pedestrian Walking with Rhythm 2.5 Normal Velocity [m/s] BPM Normal BPM Slow Rhythm Beet per Minutes (BPM)

20 rr ii = 1.8 [m] rr oo = 2.3 [m] Normal v.s. 70 BPM (Number = 6, Density = 0.47 [1/m]) Normal 70 BPM

21 rr ii = 1.8 [m] rr oo = 2.3 [m] Normal v.s. 70 BPM (Number = 24, Density = 1.86 [1/m]) Normal 70 BPM

22 Experimental and Theoretical Results Crossing Normal 70 BPM Slow rhythm improves the flow in the high-density regime.

23 Summary Pedestrian Outflow and Obstacle Conflicts and Turning affects pedestrian outflow. Appropriately set obstacle may increase pedestrian outflow. Phys. Rev. E, 76(6), , 2007 Phys. Rev. E, 80(3), , 2009 SICE Journal of Control, Measurement, and System Integration, 3(6), pp , 2010 Walking with Slow Rhythm Slow rhythm improves pedestrian flow in highdensity rhythm in a safe way. Velocity variance decreases pedestrian flow. Phys. Rev. E, 85(1), , 2012 Pedestrian and Evacuation Dynamics 2012, pp , 2014.