Title slide University of Central Florida Institute for Simulation & Training Continuous time-space simulations of pedestrian crowd behavior T.I. Lakoba, D.J. Kaup, and N.M. Finkelstein* * with Simulation Technology Center, Orlando, FL Acknowledgement: Research supported in part by STRICOM Contract # N61339-02-C-0107
Two main types of crowd models: Cellular Automata (discrete space) models : - A. Schadschneider et al (Koln, Germany); - V. Blue, J. Adler(DOT, USA) ; - J. Dijkstra et al (Eindhoven, Netherlands); - M. Batty et al (CASA @ UCL, UK); - M. Schreckenberg et al (Duisburg, Germany); Continuous-space models: - D. Helbing et al (Dresden, Germany) [social-forces + physical forces]; - S. AlGadhi et al (El-Riyadh, Saudi Arabia) [continuous-mechanics equations]; - S. Hoogendoorn et al (Delft, Netherlands) [specifies way-finding mechanisms]. Phenomena which these models can reproduce: - Lane formation in 2-way traffic; - Observed speed-density relation; - Clogging/arching at doors; - Periodic change of direction when two crowds try to pass through the same door in two opposite directions. Overview of recent work
Objectives of this work Social forces physical forces (repulsion/attraction) (pushing, friction) + • We build upon Helbing et al ’s social-force model: • To quantitatively correctly reproduce collective behavior, they assumed unrealistic parameters for individual behavior: too short an interaction range, => too high deceleration/acceleration of individual pedestrians. • We find values of parameters for Helbing’s model that correctly reproduce both collective and individual behaviors.
Outline of presentation • Describe equations of the model • Motivate need for new parameter values for the model • Highlight new features compared to Helbing’s model • The equations are numerically stiff • We propose an original algorithm that partially overcomes stiffness while using an explicit first-order Euler method • Show “movies” of pedestrians exiting a room
Equations of the model Social forces Physical forces (repulsion/attraction, (pushing, friction) preferred velocity) + Achieves or not his “walking goal” => loses/gains excitement; Has/has not seem exit/obstacle recently => gains/loses memory; Recognizes how dense crowd is => adjusts repulsion to density. }
Equations of the model:Social forces Tendency to keep preferred speed Repulsion (tendency to keep distance from others, and from boundaries) Attraction to exit(s) D attr >> D rep (non-infinite D attr plays role when a person decides which exit to head) As panic increases,
Equations of the model:Physical forces Pushing and Friction (when pedestrians come in contact with each other) • Note: • Physical forces do not depend on • relative orientation of pedestrians; • - By themselves, the pushing forces • do NOT prevent pedestrians from • “walking through” each other !
Helbing’s et al parameter values for the model m/s (normal walking) m/s (moderate panic) m/s (extreme panic) m = 80 kg, = 0.5 s, Helbing et al: [Nature, 407, p.487 (2000)] N m kg/s2 ! Yet, results of simulations, found athttp://angle.elte.hu/~panic, showremarkably realisticdynamics of many ( 200) pedestrians.
Desired parameter values • Find that lead to accelerations of no more than 0.3 – 0.5 g when considering few pedestrians. • What are the ranges of corresponding parameters? • Expect that the model needs to be made more complex to include more features that help reflect realistic human behavior. • What are the other features needed ?
Ranges for parameters:Criteria for few-ped dynamics • is found by considering a fit to measurements: • => • = free parameter found by scanning. new to • Helbing’s model
Ranges for parameters:Criterion for multi-ped dynamics The model should reproduce the “faster is slower” effect. The “faster is slower” effect: People trying to leave a room “too fast” get stuck at the door and end up getting out slower than they would have been able to do if they had walked with a “normal” speed.
Essential new featurescompared to Helbing et al’s model • Equations are stiff Code has to resolve two disparate scales: • LARGE: distances about the size of the room ( ~ 10 m), and • Small : distance between peds when they come into contact ( ~ 1 cm). New algorithm detects and eliminates overlaps among pedestrians. This allows one to keep bounded from below while using the explicit 1st-order Euler method. • Ability to learn and forget about location of an exit and walls. The knowledge about their locations is used to determine: • Direction to which a pedestrian is looking, => • Attraction force to the exit (similarly, repulsion from walls).
Overlap-eliminating algorithm • Find a pedestrian who is overlapped with, the most. • If he is overlapped with a wall, 2a.Move him away from wall so that 2b.Then, move pedestrians overlapping with him, away, and set their new velocities to coincide. • If he is overlapped, but not with a wall, do Part 2b only. • Repeat steps 1 – 3 until no overlapped pedestrians are found, but no more times than the total number of pedestrians. • Time spent on one round of O-E: During this time, coordinates of a pedestrian who is being “un-overlapped”, are not updated (i.e. he is “preoccupied” with overlap-elimination only). free parameter
Results • Simulations show presence of the “faster is slower effect”. Results are obtained as a function of parameters characterizing magnitude of the repulsive force among pedestrians. Solid: Vpref=1.5 m/s Dashed: Vpref=3 m/s Dotted: Vpref=4.5m/s
View Video of the Simulation • The video runs 3 different velocities, of one minute each. • Illustrates that “faster is slower”. • Shade of blue indicates excitement level. • Click here to watch the video.