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Crowd dynamics
• Understanding and predicting crowd behavior: central & multidisciplinary issue today
• Civil structures design − Simulating evacuations better than trying…
• Highly complex “active” dynamics − Highly stochastic non-linear dynamics, emergent
behaviors
• Numerous mathematical models have been proposed
PAGE 1 09/04/15 / CASA, Department of Mathematics and Computer Science
Driving questions
Reliable models to simulate quantitatively crowds?
Right quantitative perspective?
/ CASA, Department of Mathematics and Computer Science PAGE 2 09/04/15
Driving questions
Reliable models to simulate quantitatively crowds?
Right quantitative perspective?
/ CASA, Department of Mathematics and Computer Science PAGE 3 09/04/15
Quantitative modeling: state of the art
PAGE 4 09/04/15
Crowds: analyzed and modeled on the basis of
• Limited detailed data • ~Lab. experiments
• OR few average descriptors (egress times,vel,flux,…)
What about statistical features?
• Averages, fluctuations, rare events? • Can’t be found in limited datasets
Quantitative modeling: state of the art
Crowds: analyzed and modeled on the basis of..
• Limited detailed data • ~Lab. experiments
• OR few average descriptors (vel,flux,…)
Dynamics is richer!
• Averages + fluctuations + rare events?
/ CASA, Department of Mathematics and Computer Science PAGE 5 09/04/15
Quantitative modeling
PAGE 6 09/04/15
Capturing & reproducing statistical features
• Requirement: extensive data • Thousands exp. trajectories
• Absent up to now (impossible in lab.)
/ CASA, Department of Mathematics and Computer Science
Few tracks
PDF v,x,a,..
Checklist: Statistical dataset
Tracking technology for real world conditions
Model
PAGE 7 09/04/15 / CASA, Department of Mathematics and Computer Science
Building our statistical dataset (Metaforum Building, TU/e)
PAGE 8 09/04/15
5.2m
1.2m
1.2m
/ CASA, Department of Mathematics and Computer Science
Building our statistical dataset (Metaforum Building, TU/e)
PAGE 9 09/04/15
5.2m
1.2m
1.2m
/ CASA, Department of Mathematics and Computer Science
• 3D range sensor • 100E!/sensor • No privacy issues
Detection technology:
Kinect depth maps
PAGE 10 09/04/15
Dep
th s
cale
/ CASA, Department of Mathematics and Computer Science
5.2m
1.2m
1.2m
Pedestrians detection & tracking in brief
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1
2
3
Foreground clusterization
Depth based head detection
Head tracking
[Seer et al. 2014, Willneff et Al. 2002, Willneff 2003]
Typical dynamics
PAGE 12 09/04/15
Dataset specs: • 108 days of continuous
recording
• ~250K trajectories collected • ~2.2K traj/day
• Multiple, heterogeneous, traffic scenarios
• “Undisturbed” pedestrians
• Multiple pedestrians • “Co-flows” • “Counter-flows”
/ CASA, Department of Mathematics and Computer Science
Checklist:
Statistical dataset Tracking technology for real world conditions
Model Average single motion + U-turns Motion of pairs
PAGE 13 09/04/15 / CASA, Department of Mathematics and Computer Science
✔ ✔
“Undisturbed” pedestrians: first building block of the dynamics
2L case
Two Classes Pedestrians alone
along entire trajectory
2L (entering from
right)
2R (entering from
left)
/ CASA, Department of Mathematics and Computer Science
“Undisturbed” pedestrians: first building block of the dynamics
PAGE 15 09/04/15
• Similar dynamics • relative right • Climbing à Slower
• Fluctuations
around “crossing” pattern
• Rare Events
/ CASA, Department of Mathematics and Computer Science
“Undisturbed” pedestrians: first building block of the dynamics
/ name of department PAGE 16 09/04/15
• Similar dynamics • relative right • Climbing à Slower
• Fluctuations
around “crossing” pattern
• Rare Events
Statistical features: velocity distributions
PAGE 17 09/04/15
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 direction
Tran
sver
sal
dire
ctio
n • Consistent behavior 2L-2R • Gaussian around the mean
(u~1m/s, v~0m/s) ~ thermalized gas particles
• Rare events à rich longitudinal distrb.
2L 2R
Pedestrians as Active Brownian Particles!
PAGE 18 09/04/15 / CASA, Department of Mathematics and Computer Science
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Langevin Equation:
Effective Velocity Potential
Spatial Potential
x
y
x
y
Pedestrians as Active Brownian Particles!
PAGE 19 09/04/15 / CASA, Department of Mathematics and Computer Science
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Transversal confinement • Viscous dissipation • Harmonic potential
x
y
x
y
Langevin Equation:
Longitudinal Rayleigh-Helmoltz dynamics
PAGE 20 09/04/15 / CASA, Department of Mathematics and Computer Science
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2x
y
x
y
Longitudinal Rayleigh-Helmoltz dynamics
PAGE 21 09/04/15 / CASA, Department of Mathematics and Computer Science
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Stable velocity states
Dissipation D
issi
patio
n
Self-Propulsion
PAGE 22 09/04/15 / CASA, Department of Mathematics and Computer Science
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Stable velocity states
Random force à U-turns
Longitudinal Rayleigh-Helmoltz dynamics
PAGE 23 09/04/15 / CASA, Department of Mathematics and Computer Science
10−5
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
longitudinal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
longitudinal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−1 −0.5 0 0.5 1
transversal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−1 −0.5 0 0.5 1
transversal velocity [m/s]
Wτ (d) 2RU (s) 2R
Wτ (d) 2LU (s) 2L
Wn (d) 2RV (s) 2R
Wn (d) 2LV (s) 2L
x = v
v = �rv
K(v)�rx
V (x) + W
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Longitudinal Rayleigh-Helmoltz dynamics
Particle sim (b) vs. Data (r)
Statistics of velocity and positions: captured & reproduced
PAGE 24 09/04/15
10−5
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
longitudinal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
longitudinal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−1 −0.5 0 0.5 1
transversal velocity [m/s]
10−5
10−4
10−3
10−2
10−1
100
101
−1 −0.5 0 0.5 1
transversal velocity [m/s]
Wτ (d) 2RU (s) 2R
Wτ (d) 2LU (s) 2L
Wn (d) 2RV (s) 2R
Wn (d) 2LV (s) 2L
/ CASA, Department of Mathematics and Computer Science
10−5
10−4
10−3
10−2
10−1
100
101
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
transversal displacement [m]
10−5
10−4
10−3
10−2
10−1
100
101
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
transversal displacement [m]
10−5
10−4
10−3
10−2
10−1
100
101
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
transversal displacement [m]
10−5
10−4
10−3
10−2
10−1
100
101
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
transversal displacement [m]
∆Y (d) 2RY (s) 2R
∆Y (d) 2LY (s) 2L
∆Y (d) 2RY (s) 2R
∆Y (d) 2LY (s) 2L
Longitudinal velocity
Transversal velocity
Chord-wise confinement
Checklist:
Statistical dataset Tracking technology for real world conditions
Model Average single motion + U-turns Motion of pairs?
PAGE 25 09/04/15 / CASA, Department of Mathematics and Computer Science
✔ ✔
✔
Second building block: pair interactions!
PAGE 26 09/04/15
�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
]
Pid: 825, 826
/ CASA, Department of Mathematics and Computer Science
Two pedestrians dynamics ! perturbation of single ped. x = v
v = �rv
K(v)�rx
V (x) + W+ ✏F(x0 � x)
K(u, v) = ↵(u2 � u2p)
2 + �v2
V (y) = �y2
Conclusions • Understanding & modeling the statistic
features of pedestrian dynamics can be a step toward better quantitative crowd models. • We built a large statistical dataset • We derived a model able to
reproduce statistics in simple conditions.
• As a by-product: • automatic pedestrian tracking tool − Eindhoven Station − New light-crowd interaction
experiment in MF Market Hall
PAGE 27 09/04/15
