Understand one-dimensional microscopic models : the Follow-the-Leader model¶

The Follow-the-Leader (FTL) model presented here is a one-dimensional microscopic model. Pedestrians are assumed to walk on a line toward a common direction and the instantaneous velocity of an individual is a function of the distance to the next individual.

We consider \(N+1\) individuals walking on a straight line in a non-periodic case or in a periodic case which means that the individual \(N+1\) is identified to the individual \(1\).

We can also consider an inertial version of the Follow-the-Leader model (an order \(2\) version) which can be seen as a one-dimensional version of the Social Force model with asymmetric forcing terms to take into account that pedestrians are influenced by their immediate neighbor in front of them.

Reference : [MF2018] Chapter 2.

Thanks to several json files given as inputs of the following script, we can combine periodic or non-periodic domain with inertial or non-inertial FTL model. Examples can be find in the directory

cromosim/examples/follow_the_leader/

To run the non-inertial FTL model in non-periodic case:

python3 follow_the_leader.py --json input_ftl_order1.json

Results:

To run the non-inertial FTL model in periodic case:

python3 follow_the_leader.py --json input_ftl_order1_periodic.json

Results:

To run the inertial FTL model in non-periodic case:

python3 follow_the_leader.py --json input_ftl_order2.json

Results:

To run the inertial FTL model in periodic case:

python3 follow_the_leader.py --json input_ftl_order2_periodic.json

Results:

Code:

cromosim/examples/follow_the_leader/follow_the_leader.py

# Authors:
#     Sylvain Faure <sylvain.faure@math.u-psud.fr>
#     Bertrand Maury <bertrand.maury@math.u-psud.fr>
#
#     cromosim/examples/follow_the_leader/follow_the_leader.py
#     python follow_the_leader.py --json input_ftl_order1.json
#     python follow_the_leader.py --json input_ftl_order1_periodic.json
#     python follow_the_leader.py --json input_ftl_order2.json
#     python follow_the_leader.py --json input_ftl_order2_periodic.json
#
# License: GPL

import sys
import os
from optparse import OptionParser
import json
import scipy as sp
import matplotlib
import matplotlib.pyplot as plt
import cromosim.ftl as ftl
plt.ion()

"""
 Follow The Leader model
 Models :
 - "ftl_order1"
 - "ftl_order2" with inertia
 Domain :
 - periodic
 - non-periodic
"""

"""
 Read the input JSON file
"""

parser = OptionParser(usage="usage: %prog [options] filename",version="%prog 1.0")
parser.add_option('--json',dest="jsonfilename",
                  default="input.json",
                  type="string",
                  action="store",
                  help="Input json filename")
opt, remainder = parser.parse_args()
with open(opt.jsonfilename) as json_file:
    input = json.load(json_file)

"""
 Parameters obtained from the json file
"""

# Prefix for the result path
model = input["model"]
# tau :
if (model=="ftl_order_2"):
    tau = input["tau"]
prefix = input["prefix"]
if not os.path.exists(prefix):
    os.makedirs(prefix)
# Number of persons : 0, 1, ..., N-1
N = input["N"]
# Length of the way [0,L]
L = input["L"]
# Periodicity : True or False
# If True : no Leader
# If False : the last person (number N-1) is the leader
periodicity = input["periodicity"]
# Final time
T = input["T"]
# Timestep
dt = input["dt"]
# The positions of the persons are drawn every "drawper" iterations
drawper = input["drawper"]
# Prescribed velocity for the Leader (only used in the non-periodic case)
if (periodicity==False):
    V_leader = lambda t: eval(input["V_leader"])
# Speed function
Phi = lambda w: eval(input["speed_function"])
# To initialize the positions of the persons (regularly):
xmin_t0 = input["xmin_t0"]
xmax_t0 = input["xmax_t0"]

# To build tgrid, Nt is the number of time iterations
Nt = int(sp.floor(T/dt))+1
Nt += (Nt*dt<T)
tgrid = dt*sp.arange(Nt)
tgrid[-1] = min(T,tgrid[-1])

# data : array where the positions and the velocities will be stored
data = sp.zeros((N,2,Nt))

# Iteration counter
counter = 0

# Initialization
time = 0.0
Xold = sp.linspace(xmin_t0, xmax_t0, N)
Vold = sp.zeros(Xold.shape)
data[:,0,counter] = Xold
shift = sp.zeros(Xold.shape)


# Time loop
while (time < T-0.5*dt):
    dt = min(dt,T-time)
    if (model=="ftl_order_1"):
        if periodicity:
            X, V = ftl.update_positions_ftl_order_1(L, Xold, time, dt, Phi)
        else:
            X, V = ftl.update_positions_ftl_order_1(L, Xold, time, dt, Phi,
                                V_leader=V_leader, periodicity=periodicity)
    elif (model=="ftl_order_2"):
        if periodicity:
            X, V = ftl.update_positions_ftl_order_2(L, Xold, Vold, time, dt,
                                                    tau, Phi)
        else:
            X, V = ftl.update_positions_ftl_order_2(L, Xold, Vold, time, dt,
                                tau, Phi, V_leader=V_leader,
                                periodicity=periodicity)
    else:
        print("Bad model name... EXIT")
        sys.exit()
    ind = sp.where((X-Xold<0))[0]
    shift[ind] += L
    data[:,0,counter] = X + shift
    data[:,1,counter] = V
    time += dt
    Xold = X
    Vold = V
    counter += 1
    if (counter%drawper == 0):
        if periodicity:
            ftl.plot_people(L, X, time, data, tgrid, speed_fct=Phi,
                            ifig=10, savefig=True,
                            filename=prefix+"fig_"+str(counter).zfill(6)+".png")
        else:
            ftl.plot_people(L, X, time, data, tgrid, V_leader=V_leader,
                            periodicity=periodicity, speed_fct=Phi, ifig=10,
                            savefig=True,
                            filename=prefix+"fig_"+str(counter).zfill(6)+".png")
        plt.pause(0.01)
        print("==> Time : ",time," s, dt = ",dt," counter = ",counter)

plt.ioff()
plt.show()