Jamming phenomena of self-driven particles Pedestrian Outflow and Obstacle Walking with Slow Rhythm Daichi Yanagisawa, RCAST, UTokyo
Pedestrian Outflow and Obstacle Phys. Rev. E, 76(6), 061117, 2007 Phys. Rev. E, 80(3), 036110, 2009 SICE Journal of Control, Measurement, and System Integration, 3(6), pp. 395-401, 2010
Pedestrian Outflow (POF) 1[m] Time = POF= 2 0.5 [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?
2 1 E0 1 2 5 pp 2 2 2 2+ 1 Floor Field Cellular Automaton 2 1 2 5 pp 5 Model for Evacuation Time step = 2 5 O O O 0 2 2 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) 2 2+ 2 2 2+ 3 2 2+ 4 2 2+ 3 2 2+ 2 Only one pedestrian can enter in one cell.
Exit Simulation Does not important Important!
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
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
Schematic View of the Experiments 0 150 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 6 150 cm 2 or 3 times each (G) n=4 (H) (I)
Experiments
Theory v.s. Experiment Zipper merging (Almost 1 lane)
Quiz Which is the fastest? Exit Normal nn = 44 Shifted Obstacle nn = 44 33 1 line nn = 11 Center Obstacle nn = 55 44
Experiment Normal POF=2.78 Shifted Obstacle POF=2.92
Walking with Slow Rhythm h b r i r o N pedestrians Phys. Rev. E, 85(1), 016111, 2012. PED 2012, pp. 1291-1303, 2014.
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, 2007. Rhythm can control walking velocity of single pedestrian. What happens in a crowd case?
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.
Stride Function S and Pace Function P Stride S, Pace P 2.0 1.5 1.0 0.5 Stride S Pace P 0.0 0.0 0.2 0.4 0.6 0.8 1.0 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
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 2.0 1.5 1.0 0.5 2.0 1.5 1.0 0.5 Stride S Pace P 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Density persons m ρρ [person/m] Stride S Pace P 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Density persons m ρρ [person/m]
persons sec Flow QQ [persons/sec] Flow Q 0.8 0.6 0.4 0.2 Theoretical Analysis Convex Downward Density persons m ρρ [person/m] Fast Rhythm Normal Slow Rhythm 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (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.
Single Pedestrian Walking with Rhythm 2.5 Normal Velocity [m/s] 2 1.5 1 0.5 BPM Normal 70 0 50 100 150 200 250 BPM Slow Rhythm Beet per Minutes (BPM)
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
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
Experimental and Theoretical Results Crossing Normal 70 BPM Slow rhythm improves the flow in the high-density regime.
Summary Pedestrian Outflow and Obstacle Conflicts and Turning affects pedestrian outflow. Appropriately set obstacle may increase pedestrian outflow. Phys. Rev. E, 76(6), 061117, 2007 Phys. Rev. E, 80(3), 036110, 2009 SICE Journal of Control, Measurement, and System Integration, 3(6), pp. 395-401, 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), 016111, 2012 Pedestrian and Evacuation Dynamics 2012, pp. 1291-1303, 2014.