Local Dynamics Models for Crowd Simulation

Local Dynamics Models for Crowd Simulation Yeh, Hengchin

Outline • Introduction • Optimal Velocity Model • Helbing’s Model and Extensions • Rule Based and Others • HiDAC in More Detail • References

Introduction– Definition • Narrow • Helbing’s social forces.

Introduction– Definition • Narrow • Helbing’s social forces. • Broad • Forces • Change of positions an velocities • According to local environment • Everything not Global (planning, navigation and so on)

Introduction– Design flow • Observation • Choose the macroscopic phenomena you want to reproduce.

Introduction– Design flow • Observation • Choose the macroscopic phenomena you want to reproduce. • Design the form of (microscopic) forces. • Highly arbitrary and heuristic. • Analogous to physics.

Introduction– Design flow • Observation • Choose the macroscopic phenomena you want to reproduce. • Design the form of (microscopic) forces. • Highly arbitrary and heuristic. • Analogous to physics. • Simulation • Fix the problems.

Introduction - Examples • Domain • Roadmaps • Cellular automata • Continuous space • etc. • Methods • Particle dynamics and potential field • Rule based, eg. flocking • Special, eg. RVO. CA: Very popular in Statistical Physics (eg. Physica A), but not in Graphics

1D-Optimal Velocity Model (OVM) • From Transportation Science • 1D traffic flow. • Imaging driving on highway: • A car will keep the maximum speed with enough the distance to the next car. • A car tries to run with optimal velocity determined by the distance to the next car. • Safety distance

1D-OVM • Formula • a: How “fast” the car wants to accelerate to the desired speed. • V: optimal (desired) speed. • b, c: constants Distance to the next car • tanh? Any monotonic increasing function with upper/lower bounds suffice.

1D-OVM • Demo • Phenomena: Congestion, phase transition • The uniform flow becomes unstable when a < 2 V(L/N). • Intuition: lag in response time magnifies fluctuations.

2D-OVM • Similar ideas • For • Only attraction • For • Both attraction and repulsion θ

2D-OVM • _ • Anything that models (approximates) the anisotropic nature of human perception / reaction. • For example: • Self driving force • Range of consideration θ

2D-OVM • Similar phenomena • Lane formation in low density. • Congestion in high density

Helbing’s Forces • [Helbing and Molnar 1995] • Social force model for pedestrian dynamics • [Helbing et al. 2000] • “The paper”, published in Nature. • [Helbing et al. 2002] • [Lakoba and Kaup 2005] • [Helbing et al. 2005] • [Helbing et al. 2007] • Crowd turbulence: the physics of crowd disasters • [Yu and Johansson 2007] • Modeling Crowd Turbulence by Many-Particle Simulation

Helbing 2000 • Main equation • Add features or modify this equation. • Example: • HiDAC • AERO

Self-Driven Force • First term • Deviation of current velocity from preferred velocity • p: panic parameter; (in)dependence • : preferred velocity; “own” velocity. • : average velocity within a radius around the agent himself; “collective” velocity.

Self-Driven Force • First term • Deviation of current velocity from preferred velocity • p: panic parameter; (in)dependence • : preferred velocity; “own” velocity. • : average velocity within a radius around the agent himself; “collective” velocity. • Compared to OVM • No distance dependence for preferred velocity. • No concept of safety distance. Can be added.

Interactive Forces • Second Term

Interactive Forces • Social force: • Baseline, almost in every paper • A, B, dij

Interactive Forces • Pushing force: • Kernel • k, elasticity, spring constant

Interactive Forces • Frictional force – relative velocity • But no static friction, alternative

Agent-Obstacle Force • Analogously • Or, again

Summary of Helbing 2000 • The social force do not have a physical source. • Body force and sliding friction forces do. • But rather simple • no ground friction • no dynamic constraint • Details; Qualitative vs quantitative.

Summary of Helbing 2000 • Phenomena • Nick talked about them • Pressure buildup (Pressure discussed later) • Clogging at bottleneck • Jamming at widening • Faster is slower • Inefficient use of alternative exits (due to panicking and herding)

Lakoba and Kaup 2005 • Title: Modifications of the Helbing-Molnár-Farkas-Vicsek Social Force Model for Pedestrian Evolution • HMFV later on. • Fix some counterintuitive results of HMFV, • by changing numerical values; • as well as modifying the model

Problem 1 of HMFV • Overlapping: • HMFV allows overlapping, it NEEDS overlapping for pushing forces and frictional forces. • But no limit.

Overlapping • There should be a “core” which is not penetratable.

Overlapping • There should be a “core” which is not penetratable. • Maximum overlapping or squeezing • Smax, say, 20 % of the radius. • Collision Elimination

Methods for Handling Overlapping • HMFV: use high k in • Problem: In order to prevent overlapping  makes humans very stiff springs or bouncy balls. • 5cm  5000 N, or ~ 7G • Potential Barrier: • for • approaches infinity as dij Rij – 2 Smax

Potential Barrier • Numerically, Stiff equation • Since • f unbounded, very large. • In order to be stable, (i.e. x not “blowing off”) only very small time step can be used. • Runs forever. • Implicit integration is expensive too • Lakoba & Kaup – OEA

Overlap-Eliminating Algorithm (OEA) • n= total number of pedestrians; count = 0; • While (overlapping occurs && count < n) • Find the most overlapped pedestrian pi. • If (pi intersects with the wall) • Move pi away from the wall • set vi,n 0; vi ,t stays the same. • make pi “stationary” • end if • Move all pj’s away from pi. • Set vj vi • end while

OEA • Set vj vi this only works for uni-goal system, such as egress. • Still no guarantee. • But probability very low. • Can we do better? • What if the only collision free configuration is a “packing” one? • Finite packing? HARD

OEA time step • Determine the maximum allowable time step by letting each pedestrian to move • No less than Smax. • Can be even bigger if all (obstacles and pedestrians) are at least d > Smax apart.

OEA time step • OEA is a physical process. Need time. • Deduce the needed time from change of momentum and feasible “force”. time left for other physical processes

fOE • fOE • a free parameter, how hard he can bounce away from overlapping objects. • Related to skeleton elasticity, c.f. k for “muscle” elasticity.

Problem 2 of HMFV • Too small B • 8 cm ~ 1.4G • or say, 50 cm for less then a weight of a baseball. • Consider walking toward a wall. • Too bouncy. • Oscillation expected.

Density Effects of the Social Force • Since B is larger now • need to suppress the social repulsion as the person approaches a dense crowd density is high. K0 =0.3 K1 >1 Normalized density D0 diameter of pedestrian

Orientational Dependence of the Social Force • Face-to-back: W1 • Give extra weight to Face-to-face: W2

Helbing ‘05 • Add some more features • Impatience • : average speed into the desired direction of motion. • Long waiting times decrease the actual velocity compared to the desired one, which increases the desired velocity

More features • Fluctuation • Orientational effects 0.2 0.8

[Helbing ‘05] Interesting Suggestions

What is left in this lecture • Examples of method-specific local dynamics • in AERO • in Autonomous Pedestrians [Shao and Terzopoulos ‘05] • HiDAC in more details • following Nick’s lecture.

Local Dynamics in AERO

Local Dynamics in AERO • New face: Roadmap force field lk p

Autonomous Pedestrians • Rule-based. • local rules • A B D E F: collision avoidance. • C Modified Potential field. To maintain separation in a moving crowd.

Autonomous Pedestrians • Temporary Crowd: • Moving in similar directions • Situated within a parabolic region in front of H. • ri repulsiveness • di distance to Ci fi di

HiDAC • Position for agent i is

Local Dynamics Models for Crowd Simulation