Moving in a crowd: human perception as a multiscale process

0.45 MB·English

by  Annachiara Colombi

MOVING IN A CROWD: HUMAN PERCEPTION AS A
MULTISCALE PROCESS
ANNACHIARACOLOMBI,MARCOSCIANNA,ANDANDREATOSIN
Abstract. The strategic behaviour of pedestrians is largely determined by
how they perceive and react to neighbouring people. This issue is addressed
5
in this paper by a model which combines, in a time and space-dependent
1
way, discrete and continuous effects of pedestrian interactions. Numerical
0
simulations and qualitative analysis suggest that human perception, and its
2
impactoncrowddynamics,canbeeffectivelymodelledasamultiscaleprocess

c basedonadualmicroscopic/macroscopicrepresentationofgroupsofagents.
e
D

3
1. Introduction and motivations

]
h In this paper we aim at incorporating the effect of pedestrian perception in a
p mathematical description of interpersonal interactions.
- We take inspiration from [3, 4], where the author points out that different per-
c
o ceptions of the surroundings can lead walkers to react in a more individualistic or
s group-oriented way to the presence of nearby people. In particular, he introduces
.
s the concept of the use of space as an indicator of such a behaviour, implying that
c
this affects the pedestrian collision avoidance mechanism.
i
s We start by the celebrated social force model [6] in the simple case of a sin-
y
gle moving walker and we enrich it by introducing a multiscale micro/macroscopic
h
representation of a group of individuals composing a static crowd that the walker
p
[ interacts with. For this, we take advantage of the measure-theoretic multiscale ap-

proach developed in [2]. The multiscale representation uses a perception function,
2
to be related to the aforesaid use of space, which determines how much the inter-
v
5 actions of the walker are directed towards either the individual (viz. microscopic)
7 or the collective (viz. macroscopic) distribution of the nearby people.
3 Our results show that such a multiscale interpretation of the effect of human
1
perceptioncangreatlyimpactonthecorrectreproductionofpedestriantrajectories
0
and that this may not be equally possible with a single-scale model.
.
2
0 2. Mathematical model
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1 Weconsiderforsimplicityasinglepedestrianinatwo-dimensionaldomain, who
: walksthroughastaticcrowdtoreachagiventarget. Thepedestrianisrepresented
v
i by his/her position and velocity x(t), v(t) ∈ R2, respectively, where t ≥ 0 is time.
X
The distribution of the static individuals is instead described by a Radon positive
r measure µ carrying a total mass µ(R2)=N, i.e., the number of static individuals.
a
The dynamics of the walker are ruled by a social-force-type model [6]:
(cid:18) (cid:19)
|w(t)|
(1a) x˙(t)=v(t)=g w(t)
v
max
v (x(t))−v(t) (cid:90)
(1b) w˙(t)= d + K(y−x(t))dµ(y),
τ
Sα(x(t))
R
2010 Mathematics Subject Classification. 37N99,82C22,90B20.
Keywordsandphrases. Pedestrianperception,useofspace,multiscalemodel,measuretheory.
1
2 ANNACHIARACOLOMBI,MARCOSCIANNA,ANDANDREATOSIN
where g(z)=min{1, 1} bounds the actual speed |v(t)| by a physiological maximal
z
value v >0.
max
In (1b), v : R2 → R2 is a given desired velocity representing the preferred di-
d
rection of the moving pedestrian to reach his/her destination from his/her current
position, and τ is a relaxation time. The second term at the right-hand side mod-
els instead the interactions with the static individuals. In particular, it expresses
the tendency to keep a sufficient distance from them for collision avoidance. The
interaction kernel K : R2 → R2 describes the position-dependent repulsion of the
walker from the static individuals within in his/her sensory region Sα(x(t)) (see
R
Fig. 1, bottom-left panel).
2.1. Modelling perception: the multiscale structure of µ. Themeasureµis
usedtodescribehowthestaticindividualsareperceivedbythemovingpedestrian,
who can interact with them either singularly or group-wise depending on his/her
use of space (see Fig. 1 top-left panel).
Takinginspirationfrom[1,2],weassumethatahighlylocalisedperception,typi-
calofrelaxedconditions,inducesaquiteaccurateuseofspace,henceindividualistic
interactions. Inthiscasewechooseµasanatomicmassmeasureµ=(cid:15):=(cid:80)N δ ,
k=1 yk
where δ is the Dirac delta and y ∈R2 is the position of the kth static individual.
k
Conversely, a blurred perception, typical of hurried or panicky conditions, induces
a rougher assessment of the usable space, hence group-oriented interactions. In
thiscasewechooseµasaLebesgue-absolutelycontinuousmeasureµ=ρL2,where
ρ : R2 → [0, +∞) is the density of the static crowd. With a slight abuse of nota-
tion, we will denote the measure µ by the same symbol ρ and we will require that
ρ(R2)=(cid:82) ρ(y)dy =N.
R2
Themovingpedestriancanalsochangehis/hertypeofperceptionwhilewalking,
for instance according to local characteristics of the static crowd. We model this
by generalising µ as
(cid:0) (cid:1)
µ =θ(x(t))(cid:15)+ 1−θ(x(t)) ρ,
t
whereθ :R2 →[0, 1]isthelevelofperception. θ (cid:38)0indicatesablurredperception,
with the moving pedestrian tending to assess the space occupancy in a continuous
way. Conversely,θ (cid:37)1indicatesalocalisedperception,withthemovingpedestrian
tending to assess it in a discrete way. Note that the dependence of θ on x(t) makes
the measure µ time-dependent.
3. Numerical simulations
Weperformnumericalsimulationsofmodel(1a)-(1b)inatwo-dimensionalrect-
angular domain of size 4 m×10 m, which is meant to reproduce a corridor or a
pavement.
The moving pedestrian, initially in x(0)=(1.25, 1) m, wants to reach a 85 cm-
wide target on the top edge of the domain. In doing this, s/he faces N = 8 static
individuals. Weconsidertwospatialarrangementsofthelatter: (i)theyaresparse;
(ii) they form a dense cluster in the central part of the domain, see Fig. 1 right
panels. Given their microscopic positions {y }N , we construct their macroscopic
k k=1
density ρ as the superposition of N unit-mass cones:
N (cid:18) (cid:19)
(2) ρ(y)= 3 (cid:88) 1− |y−yk| χ (y)
πσ2 σ Bσ(yk)
k=1
with σ =0.5 m, χ being the characteristic function of the ball centred in y
Bσ(yk) k
with radius σ.
System (1a)-(1b) requires the specification of some parameters, see Table 1.
Moreover, we define v (x)=v xd−x(t) , where x =(1.2, 10) m is the centre of
d max|xd−x(t)| d
MOVING IN A CROWD: HUMAN PERCEPTION AS A MULTISCALE PROCESS 3
Figure 1. Top-left: pictorialrepresentationofthemultiscaleper-
ception. Bottom-left: sensoryregionandinteractionkernel. Right:
specification of settings (i), (ii)
Table 1. Summary of the parameters used in the model
Parameter Description Value Reference
v pedestrian maximum speed 1.34 m/s [5]
max
τ relaxation time 0.5 s [5]
R sensory radius 3 m [5]
α half visual angle 100◦ [5]
k interpersonal repulsion coefficient 0.3 m2/s2 tuned here
0
R average pedestrian body radius 0.3 m [7, 9]
b
the target. We set the sensory region of the moving pedestrian to be the circular
sector
(cid:26) (cid:27)
(y−x(t))·v (x(t))
Sα(x(t))= y ∈R2 :|y−x(t)|≤R, d ≥cosα ,
R v |y−x(t)|
max
whereRistheinteractionradiusandαisthehalfvisualangle. Thiscircularsector
isorientedinsuchawaythatthegazedirectionofthemovingpedestrianisaligned
with v , thus with the target (cf. Fig. 1, bottom-left panel). Finally, we take the
d
interaction kernel as a classical distance-decaying function:
 (cid:16) (cid:17)
−k0(cid:16)R1b − R1(cid:17) |rr| if 0≤|r|≤Rb
K(r)= −k 1 − 1 r if R <|r|≤R
0 0 |r| R |r| othebrwise
(cf. Fig.1bottom-leftpanel),whereR <Risthebodysizeofanaverageindividual
b
and k >0 is a proportionality coefficient.
0
Wenowperformnumericalteststoseehowdifferenttypesofperceptiongiverise
to different migratory paths of the moving pedestrian. We consider either a fully
localisedperception,givenbyθ ≡1,whichcorrespondstothegenuinelymicroscopic
social-force-type model, or a hybrid one. In this latter case, we assume that the
walkerhasalocalisedperceptionwhenthestaticindividualsinSα(x(t))aresparse
R
enough. On the contrary, when they are more densely packed s/he perceives them
4 ANNACHIARACOLOMBI,MARCOSCIANNA,ANDANDREATOSIN
Figure 2. Paths followed by the moving pedestrian in the two
simulation settings
as an undifferentiated group. The discriminating quantity is the mean distance (cid:96)
among the static individuals within the sensory region:
1 (cid:88)
(3) (cid:96)=(cid:96)(x(t))= |y −y |,
n(n−1) h k
yh,yk∈SRα(x(t))
where n=#{y ∈Sα(x(t)), k =1, ..., N} is their number. Then we set:
k R

0 if 0≤ (cid:96) ≤1
(cid:18) (cid:96) (cid:19)  (cid:96)∗
(4) θ =θ = (cid:96) −1 if 1< (cid:96) ≤2
(cid:96)∗ (cid:96)∗ (cid:96)∗
1 if (cid:96) >2
(cid:96)∗
where (cid:96)∗ =1 m is a reference value. Actually, (3) is valid only if n≥2. If instead
n=0, 1 we invariably set θ =1.
As shown in Fig. 2a, c, in both settings (i) and (ii) the fully localised perception
allows the walker to pass in between the static individuals, thereby following an
almost straight path towards the target. A hybrid variable perception results in-
steadindifferenttrajectoriesdependingonthecrowddistribution. Whenthestatic
individuals are sufficiently sparse (setting (i), Fig. 2b) the moving pedestrian still
perceivesthemasasetofsingleelementsands/heusesthefreespaceamongthem.
Conversely, when they are more densely packed (setting (ii), Fig. 2d) the moving
pedestrian perceives predominantly their ensemble as a compact distributed mass
and s/he circumnavigates the density spot. Note that, instead, the purely micro-
scopic model may not allow one to appreciate substantial differences between the
migratory paths in settings (i) and (ii) (cf. Figs. 2a, c).
4. Analysis of the trajectories
We now study the dependence of the trajectory t (cid:55)→ x(t) on the perception
function θ and on the multiscale description ((cid:15), ρ) of the static crowd. We begin
by rewriting (1b) in the compact form w˙(t)=a[µ ](x(t), w(t)), where a stands for
t
the acceleration of the moving pedestrian.
MOVING IN A CROWD: HUMAN PERCEPTION AS A MULTISCALE PROCESS 5
Assumption 4.1. We assume that a is bounded and Lipschitz continuous, i.e.,
there exist a , Lip(a)>0 s.t.:
max
|a[µ](x, w)|≤a
max
|a[ν](x , w )−a[µ](x , w )|≤Lip(a)(|x −x |+|w −w |+W (µ, ν))
2 2 1 1 2 1 2 1 1
for all x, x , x , w, w , w ∈R2 and µ, ν ∈MN(R2).
1 2 1 2 +
Remark. Here and henceforth MN(R2) is the cone of positive measures with mass
+
N in R2. Furthermore, W is the first Wasserstein metric in the space of finite
1
positive measures.
Nextweconsideranytwomeasuresµ1, µ2 ∈MN(R2)describingthedistribution
t t +
of the static crowd and we let (x (t), w (t)), (x (t), w (t)) be the corresponding
1 1 2 2
trajectory-velocity pairs of the moving pedestrian.
Proposition 4.2. Let x (0) = x (0) and w (0) = w (0). There exists a constant
1 2 1 2
C >0 such that
(cid:90) t
(5) |x (t)−x (t)|≤CeCt W (µ1, µ2)ds
2 1 1 s s
0
for all 0≤t≤T <+∞.
Proof. Integratingtheaccelerationintimeinthetwocasesandtakingthedifference
gives
(cid:90) t
|w2(t)−w1(t)|≤ (cid:12)(cid:12)a[µ2s](x2(s), w2(s))−a[µ1s](x1(s), w1(s))(cid:12)(cid:12) ds,
0
whence, by Assumption 4.1 and Gronwall’s inequality,
(cid:90) t
(6) |w (t)−w (t)|≤Lip(a)eLip(a)t (cid:0)|x (s)−x (s)|+W (µ1, µ2)(cid:1) ds.
2 1 2 1 1 s s
0
Now, integrating (1a) in time and using the boundedness and Lipschitz continuity
of g we obtain
a Lip(g)t(cid:90) t
|x (t)−x (t)|≤ max |w (s)−w (s)| ds
2 1 v 2 1
max 0
(cid:90) t
+Lip(a)t (cid:0)|x (s)−x (s)|+|w (s)−w (s)|+W (µ1, µ2)(cid:1) ds,
2 1 2 1 1 s s
0
which, invoking (6), after standard manipulations produces
(cid:18)(cid:90) t (cid:90) t (cid:19)
|x (t)−x (t)|≤α(t) |x (s)−x (s)| ds+ W (µ1, µ2)ds ,
2 1 2 1 1 s s
0 0
(cid:104) (cid:16) (cid:17) (cid:105)
with α(t) = Lip(a)t 1+ amaxLip(g) +Lip(a) teLip(a)t . Since α(t) is non-decre-
vmax
asing, we set C =α(T)≥α(t) and by Gronwall’s inequality we get the thesis. (cid:3)
It is not difficult to check that slightly regularised versions of both v , cf. [2],
d
and the acceleration in (1b) satisfy Assumption 4.1. In particular, we propose
(cid:16) (cid:17)
v (x)−g |w| w (cid:90)
a[µ](x, w)= d vmax + K(y−x)η (y)dµ(y),
τ R2 SRα(x)
where η :R2 →[0, 1] is a mollification of the characteristic function of the set
Sα(x)
R
Sα(x). To see that Assumption 4.1 is satisfied, use the boundedness and Lipschitz
R
continuity of v and g and the results contained in [8].
d
Thanks to Proposition 4.2 we are now in a position to prove
6 ANNACHIARACOLOMBI,MARCOSCIANNA,ANDANDREATOSIN
Theorem4.3. Letθ , θ :R2 →[0, 1]beLipschitzcontinuousandµi =θ (x (t))(cid:15)+
1 2 t i i
(cid:0) (cid:1)
1 − θ (x (t)) ρ, i = 1, 2, the corresponding multiscale measures. There exists
i i
C >0, which depends on emin{Lip(θ1),Lip(θ2)}W1((cid:15),ρ), such that
sup |x (t)−x (t)|≤CW ((cid:15), ρ)(cid:107)θ −θ (cid:107) .
2 1 1 2 1 ∞
t∈[0,T]
Proof. Let ϕ:R2 →R be any Lipschitz continuous function with Lip(ϕ)≤1, then
(cid:12)(cid:90) (cid:12) (cid:12)(cid:90) (cid:12)
(cid:12)(cid:12)(cid:12) R2ϕ(y)d(µ2t −µ1t)(y)(cid:12)(cid:12)(cid:12)=|θ2(x2(t))−θ1(x1(t))|·(cid:12)(cid:12)(cid:12) R2ϕ(y)d((cid:15)−ρ)(y)(cid:12)(cid:12)(cid:12).
Taking the supremum of both sides over ϕ yields
W (µ1, µ2)≤(cid:0)|θ (x (t))−θ (x (t))|+|θ (x (t))−θ (x (t))|(cid:1)W ((cid:15), ρ)
1 t t 2 2 1 2 1 2 1 1 1
(cid:0) (cid:1)
≤ (cid:107)θ −θ (cid:107) +Lip(θ )|x (t)−x (t)| W ((cid:15), ρ).
2 1 ∞ 1 2 1 1
Ananalogousresultisobtainedbyaddingandsubtractingθ (x (t)), butthisgives
2 1
Lip(θ ) before the second term at the right-hand side. Thus finally:
2
W (µ1, µ2)≤(cid:0)(cid:107)θ −θ (cid:107) +min{Lip(θ ), Lip(θ )}|x (t)−x (t)|(cid:1)W ((cid:15), ρ)
1 t t 2 1 ∞ 1 2 2 1 1
andthethesisfollowsbypluggingthisin(5)andinvokingGronwall’sinequality. (cid:3)
Theorem 4.3 supports the numerical findings of the previous section. In both
settings (i) and (ii) the purely microscopic model corresponds to θ ≡ 1, while
1
the hybrid model corresponds to θ = θ (x(t)) as indicated in (3)-(4). In general,
2 2
(cid:107)θ −θ (cid:107) = 1 as soon as θ (x) = 0 for some x ∈ R2, hence the relationship
2 1 ∞ 2
between the trajectories t (cid:55)→ x (t), x (t) depends strongly on the multiscale de-
1 2
scription of the static crowd. In setting (i) the microscopic and macroscopic distri-
butions of the static crowd are similar, because the crowd is sparse. Consequently
W ((cid:15), ρ) is small and Theorem 4.3 implies that no relevant differences can be ob-
1
served in the trajectories of the moving pedestrian. Conversely, in setting (ii) the
twodistributionsofthestaticcrowdarequitedifferentbecauseofthedensityspot.
Therefore W ((cid:15), ρ) is large and Theorem 4.3 admits possibly different trajectories
1
of the moving pedestrian.
5. Conclusions
We have proposed a mathematical model for pedestrian movement which im-
plements the idea of human perception as a multiscale process. In more detail, it
takes into account the fact that the way in which a walker perceives and reacts
to the presence of other nearby individuals changes according to various environ-
mental factors, among which we have considered especially his/her use of space.
We have modelled the perception and the consequent use of space by means of a
dual micro/macroscopic representation of the nearby individuals. Our numerical
andanalyticalresultsshowthatdifferenttypesofperceptiongreatlyimpactonthe
actual migratory paths of the walkers, which may not be reproduced by models at
a single scale.
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Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: [email protected]
Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: [email protected]
Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: [email protected]