Stas$cal(Structures(of(Low( Density(Pedestrian(Dynamics(( faculteit/Afdelingen... ·...
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Sta$s$cal Structures of Low Density Pedestrian Dynamics Alessandro Corbe:a Eindhoven University of Technology, NL with: ChungMin Lee (CSULB), Roberto Benzi (Rome 2), Adrian Muntean (TU/e), Federico Toschi (TU/e) Centre for Analysis, Scien2fic compu2ng and Applica2ons Traffic and Granular Flow ’15 October 28 th , 2015
Transcript of Stas$cal(Structures(of(Low( Density(Pedestrian(Dynamics(( faculteit/Afdelingen... ·...
Sta$s$cal Structures of Low Density Pedestrian Dynamics
Alessandro Corbe:a Eindhoven University of Technology, NL
with:
Chung-‐Min Lee (CSULB), Roberto Benzi (Rome 2), Adrian Muntean (TU/e), Federico Toschi (TU/e)
Centre for Analysis, Scien2fic compu2ng and Applica2ons
Traffic and Granular Flow ’15 October 28th, 2015
Introduc$on & Mo$va$on
• Walking pedestrians: rich & complex dynamics – Reliable models: relevant in science & technology
• Stochas$c, nearly unpredictable mo$on – Quan4ta4ve-‐(sta$s$cal) assessment of fluctua4ons? • Measurements? • Rare behaviors? • Modeling?
Introduc$on & Mo$va$on • This presenta$on: – “low density” pedestrian dynamics in a corridor
• Quan$ta$ve-‐(sta$s$cal) models?
• Content: 1. High sta$s$cs measurement approach
2. Analysis of stochas$c fluctua$ons 3. Quan$ta$ve model up to rare events
Introduc$on & Mo$va$on • This presenta$on: – “low density” pedestrian dynamics in a corridor
• Quan$ta$ve-‐(sta$s$cal) models?
• Content: 1. High sta$s$c resolu$on measurements
2. Analysis of stochas$c fluctua$ons Quan$ta$ve model up to rare events
High sta$s$cs measurements approach
• Real-‐life secng • 1y recording ~h24, • ~2.2K people every weekday • ~230K tracks dataset
Metaforum building, TU/e
[Seer et al. 2014, Corbe:a et al. 2014, Corbe:a et al. 2015]
0th floor, canteen 1st floor, dining area
Pedestrian tracking technique • 1. Heads detec2on – Overhead, 3d view
• Depthmap-‐based, Via Microsok Kinect • (Complete) clustering of “depth-‐cloud” • 15Hz sampling
(Seer et Al., 2014)
Clusteriza$on tree cut at “body scale” Heads marked as low depth (closest) percen$les
• 2. Head tracking • Head Spa$o-‐Temporal matching via 3DPTV
• from experimental fluid mechanics (Willneff et Al. 2002, Willneff 2003)
• Nearest search with velocity predic$on
• Implementa$on:
Pedestrian tracking technique
�2.5 �2.0 �1.5 �1.0 �0.5 0.0 0.5 1.0 1.5 2.0 2.5X[m]
�1.0�0.8�0.6�0.4�0.2
0.00.20.40.6
Y[m
]
Rec. win.: 2.2m x 1.2 m
Acknowledgment: A. Liberzon (TAU)
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Traffic -‐ local occupancy
Lunch peak
Break peak
week 3rd Feb ‘14
5th Feb ‘14
1 week
Occupancy = 2 Occupancy = 5
Par$$oning ensemble trajectories in flow classes => sta$s$cs per-‐class
Many flow condi$ons
Undisturbed pedestrians
Co-‐Flows
Counter-‐flows
Co-‐flow, to Lek
Co-‐flow, to Right
Counter-‐flow
Ensemble of trajs.
Fundamental diagrams
• L-‐R symmetry broken • Descending direc$on
faster • Counter-‐flow > Co-‐flow
(at same load) • Ped. ascending
might have trays
Simple per-‐class sta$s$cs on veloci$es
[Corbe:a et. al 2014]
Beyond average values…
• Full probability distribu$on func$ons
– Analyze stochas$city
– Mathema$cal models
• Now: undisturbed pedestrians
Undisturbed pedestrian dynamics
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�1.0�0.8�0.6�0.4�0.2
0.00.20.40.6
Y[m
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• Simple scenario • Pedestrians cross the corridor (L -‐> R) • No reasons to stop 30.09.2013 09:55:52.114
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Undisturbed pedestrian dynamics
• Rare events: trajectory inversions
High-‐sta$s$cs perspec$ve
�2.5 �2.0 �1.5 �1.0 �0.5 0.0 0.5 1.0 1.5 2.0 2.5
x [m]
�1.0
�0.8
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�0.2
0.0
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0.6
y[m
]
1. Preferred walking path 2. “Confined” transversal mo$on 3. longit. & transv. fluctua$ons
High-‐sta$s$cs perspec$ve
1. Preferred walking path 2. “Confined” transversal mo$on 3. longit. & transv. fluctua$ons
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−3.054 −2.054 −1.054 −0.054 0.946
longitudinal velocity Wτ [m/s]
transversal velocity Wn[m/s]
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2.973 −1.946 −0.919 0.108
longitudinal velocity Wτ [m/s]
transversal velocity Wn[m/s]
Wτ (m) 2LWn (m) 2L
Wτ (m) 2RWn (m) 2R
LONGITUDINAL
TRAN
SVER
SAL
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Can we reproduce this behavior in sta$s$cal sense?
Ac$vity (ac$ve fric$on for propulsion)
Spa$al confinement
Random external factors
Langevin-‐like equa$on Second order stochas$c dynamics:
Stochas$c mo$on around preferred path: Quadra$c poten$al for posi$on (V) and velocity (K)
Transversal fluctua$ons
v = �2�v � 2�y + �yw
10−5
10−4
10−3
10−2
10−1
100
101
−1 −0.5 0 0.5 1
wn [m/s]
measurementsmodel
10−5
10−4
10−3
10−2
10−1
100
101
102
−1 −0.5 0 0.5 1
y′ [m]
measurementsmodel
Posi$on PDF Model vs. data
Confined Gaussian fluctua$on:
Velocity PDF Model vs. data
Stable velocity states
Bi-‐stable longitudinal mo$on
4th order velocity poten$al velocity (K) Simplest bi-‐stable stochas$c velocity dynamics
Small fluctua$ons
Stable velocity states
Bi-‐stable longitudinal mo$on
Trajectory inversions
4th order velocity poten$al velocity (K) Simplest bi-‐stable stochas$c velocity dynamics
10−6
10−5
10−4
10−3
10−2
10−1
100
101
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
wτ [m/s]
measurementssimulations
Velocity PDF
• Inversion dynamics captured in velocity pdf • Rare and uncorrelated => Poisson sta2s2cs
10−3
10−2
10−1
0 200 400 600 800 1000 1200 1400
Ni
measurementsexponential E[Ni] = 450
simulations
Inversion PDF
Bi-‐stable longitudinal mo$on
u = 4↵u(u2 � u2p
) + �x
w[A. Corbe:a et. al, to be submi:ed]
Conclusion • Analyzed pedestrian dynamics via large experimental datasets
– Sta$s$c insights possible – Analogous features expected in low density crowds
• Simple Langevin-‐like model to reproduce stochas$c features of undisturbed pedestrians mo$on – Quan$ta$ve – Small fluctua$ons and rare inversions captured within same model
• Next: – Avoidance dynamics in pairs & higher order interac$ons
[A. Corbe:a, Phd Thesis, 2015 – soon online] – Sta$s$city inves$ga$on at high density regimes?
References 1. A. Corbe:a, L. Bruno, A. Muntean, F. Toschi, High Sta$s$c Measurements of
Pedestrian Dynamics, 2014, Transporta$on Research Procedia, 2. 2. S. Seer, N. Brandle and C. Rac, Kinects and human kine4cs: a new approach for
studying pedestrian behavior, Transporta$on Research Part C: Emerging Technologies, 2014, 48 : 212-‐228.
3. A. Corbe:a, A. Muntean, K. Vafayi, F. Toschi, Parameter Es$ma$on of Social Forces in Pedestrian Dynamics models via a Probabilis$c Method, 2015, Mathema$cal Biosciences and Engineering, 12 (2), 337-‐356
4. The OpenPTV ini$a$ve, 2012 -‐, www.openptv.net 5. J. Willneff, A. Gruen, A new Spa$o-‐Temporal Matching Algorithm for 3D-‐Par$cle
Tracking Velocimetry, The 9th of Interna$onal Symposium on Transport Phenomena and Rota$ng Machinery, Honolulu, Hawaii, 2002
6. J. Willneff, A Spa$o-‐Temporal Matching Algorithm for 3D Par$cle Tracking Velocimetry, PhD Thesis, ETH-‐Zurich, 2003
7. A. Corbe:a, Mul$scale crowd dynamics: physical analysis, modeling and applica$ons, PhD Thesis, 2015
8. A. Corbe:a, C. Lee, R. Benzi, A. Muntean, F. Toschi, Fluctua$ons and mean behaviours in diluted pedestrian flows, to be submi:ed
Acknowledgements: L. Bruno, A. Holten, A. Liberzon, A. Tosin, G. Oerlemans, Intelligent Ligh$ng Ins$tute (Eindhoven)
