Crowd Simulations

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Crowd Simulations

• Sashi Kumar Penta
• Robotics Class Presentation

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Overview

• Motivation
• Background
• Behavior Planning for Character Animation,
  Kuffner et.al, Symposium on Computer Animation
  2005
• Continuum Crowds, Adrein et.al, SIGGRAPH 2006
• Summary

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Motivation

• Crowds are every where!!
• Motivating video Clip
• Movies
• The Chronicles of Nornia The witch, The Lion
  and The ward robe
• The Lord of the ring
• I, Robot, etc

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Applications
Courtesy Illknur

• Entertainment industry (animation production,
  computer games)
• Training of police military (demonstrations,
  riots handling)
• Architecture (planning of buildings, towns,
  visualization)
• Safety science (evacuation of buildings, ships,
  airplanes)
• Sociology (crowd behavior)
• Transportation research for urban planning

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Background

• Motion-graph approaches
• Gleicher et. al TOG 2003
• Kuffner et. al SCA 2005
• Static potential fields
• Kirchner 2002
• Continuous density field
• Hughes 2003
• Adrien et. al SIGGRAPH 2006

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• Behavior Planning for Character Animation,
  Kuffner et.al, SCA 2005

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Abstract

• Automatically generate realistic motions for
  animated characters
• Motion clips - FSM - movements of a virtual
  character
• Global search of the FSM a sequence of
  behaviors for the character

From small amount of data
Scales to large motion DBs
Flexible framework
Static/Dynamic environments
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Behavior Finite-State Machine

• Behavior FSM defines movement capabilities of
  the character
• Each state consists of a collection of motion
  clips that represent a high-level behavior
• Each directed edge represents a possible
  transition b/w two behaviors
• There can be multiple motion clips with in a
  state.

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Environment Abstraction

• Environment as a 2D height-field gridmap
• Gridmap also encodes
• Obstacles that the character should avoid
• Free space where character can navigate
• Information about special obstacles such as an
  archway
• Height value is used so that we can represent
  terrains with slopes or hills.
• Virtual character is bounded by a cylinder with
  radius r position is the center of the cylinder

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Data structures

• A tree with nodes that record explored states in
  the FSM
• Each node in the tree stores the motion clip,
  position, orientation, time and cost
• Priority queue of FSM states ordered by cost,
  which represent potential nodes to be expanding
  during the next search iteration

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A - search

• Total cost
• ( cost of the path up to that node)
• (expected cost to reach the goal)
• Using A-search, The planner iteratively expands
  the lowest cost node in the queue until either
  the a solution is found or the queue is empty.

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ALGORITHM 1 BEHAVIOR PLANNER

• Function T takes the input state sin and an
  action a as parameters and returns the output
  state sout

• Function F returns the set of actions A that
  the character is allowed to take from sbest

• Function G determines if we should expand
  Snext as a child of sbest in the tree.

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Motion Generation Blending

• Search algorithm returns a sequence of behaviors
• Sequence is converted into an actual motion for
  the character
• Blending to smooth out any slight
  discontinuities where the transitions b/w
  behaviors occur.

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Advantages

• Scalability
• Memory Usage
• Intuitive Structure
• Generality
• Optimality
• Anytime

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Limitations

• This technique requires an existence of a
  behavior FSM
• This technique expects the data has been
  appropriately segmented categorized

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Results
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What needs to be taken care

• Individual human motion
• Environmental constraints
• Intelligent path planning

Large group of people exhibit behavior of
enormous complexity and subtlety.

• Dynamic interactions between people

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Previous methods

• Agent based methods
• Funge et al. 1999
• Shao and Terzopoulos 2005
• Massive Software 2006
• Graph based techniques
• Bayazit et al. 2002
• Kuffner et al. 2005
• Static potential fields
• Goldeinstein et al. 2001
• Continuous density field
• Hughes 2002
• Adrien et. al 2006

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Agent based methods

• Pros
• Operate with each individual making independent
  decisions
• Capture each persons unique situation
  visibility, proximity of other pedestrians, etc
• Different simulation parameter may be defined for
  each crowd member
• Cons
• Computationally expensive
• Difficult to develop behavioral rules that
  consistently produce realistic motion

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Key points

• Real-time crowd model based on continuum dynamics
• Global navigation with moving obstacles such as
  other people
• Motion of large crowds without need for explicit
  collision avoidance

Dynamic potential field
Interactive rates
Smooth flow
Emergent phenomena
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The Governing Equations

• Hypothesis 1
• Each person is trying to reach a geographic goal
  G ? R2
• Hypothesis 2
• People move at the maximum speed possible.
• Maximum speed field f
• Where
• Hypothesis 3
• There exist a discomfort field g so that, people
  would prefer to be a point x rather than x if
• g(x) gt g(x)

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Tying together Hypothesis 4

• People choose paths so as to minimize a linear
  combination of
• The length of the path
• The amount of time to the destination
• The discomfort felt, per unit time, along the
  path
• Hypothesis 4
• A path P that minimizes

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Optimal path Computation

• Suppose, Cost function
  every where equal to the cost of the optimal
  path to the goal.
• Optimal strategy to move the opposite the
  gradient of this function.
• Potential function ( ) by following set of
  all optimal paths outwards from the goal.
• In the goal 0
• Every where,

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Speed field

• Speed is a density-dependent variable
• Crowd density field
• Low density (lt min) depends on slope
• High density (gt max) depends vel of crowd
• Medium density (gt min lt max)

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Models of the future

• Predictive discomfort
• Future path planning through a constantly updated
  static view of environment.
• Expected periodic changes
• When the field deterministically changes over time

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Algorithm

• For each time step
• Convert the crowd to a density field.
• For each group
• Construct the unit cost field C.
• Construct the potential and its gradient
• Update the peoples locations
• Enforce the minimum distance between people

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Implementation details

• Physical fields as 2D arrays of floating point
  numbers.
• Scalar fields defined at the center of each
  grid cell
• Its true for average velocity ( ), stored
  as pair of floats
• Anisotropic fields those depend on both
  position and
• direction
• Stored with four floats per
• cell corresponding to
• 0o, 90o, 180o, 270o , i.e.
• EAST, NORTH, WEST and SOUTH
• faces of each cell

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Density Conversion

• Splat the crowd particles onto a density grid,
  to compute speed field which are dependent on
  density
• Requirements of density function
• Must be continuous w.r.t location of the people
  
• to avoid sharp discontinuities in density and
  subsequently to the speed
• Each person should contribute no less than
  to their own grid cell, but no more than to
  any neighboring grid cell
• to ensure each individual is not affected by its
  own contribution to the density field

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Density Conversion ctd..

• For each person, we find the closest cell center
  whose coordinates are both less than that of the
  person
• Relative coordinates of that person
  w.r.t the cell center
• Persons density then added
• to the grid as

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Unit cost field

• Compute speed field f
• Then calculate cost field C using
• f and C are anisotropic
• We would evaluate the speed
• and discomfort at the cell into
• which the person would be
• moving if they choose that direction

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Dynamic Potential Field

• Constructing is the dynamic potential is the most
  complex and time consuming step of the algorithm
• Implicit eikonal equation Solution
• Assign the potential field inside the goal to
  0 include these cells in KNOWN cells all other
  to UNKNOWN
• UNKNOWN cells adjacent to KNOWN cells are
  included in the CANDIDATE cells and we
  approximate at these locations by solving a
  finite difference approximation to
• The CANDIDATE cell with the lowest potential is
  then included in the KNOWN cells and its
  neighbors are introduced into the CANDIDATE set
  by re-approximating the potentials at these cells
• This process is repeated, propagating the KNOWN
  cells outwards from the goal until all cells are
  defined.
• HEAP DATA STRUCTURE O(N log N)

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Finite difference approximation

• First find the less costly adjacent grid cell
  along the both x- and y-axis
• Solve the equation for
• Once we have computed , we
• take its difference with the neighboring
• grid cells in the upwind direction gives us
• Renormalize the gradient, multiply by the speed
  in the appropriate directions to compute the
  velocity field at that point

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Crowd Advection

• Once velocity field is known
• Each persons position is displaced by their
  velocity, which is effectively computing an Euler
  integration to
• Min displacement Enforcement
• All pairs within a threshold distance,
  symmetrically pushed apart so that min distances
  are preserved

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General Algorithm Overview
Grids
Potential Fields
New positions
Density
Goal
Boundary
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Results
Video
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Contributions

• Integrates both global navigation local
  collision avoidance into one framework
• Velocity dependent term which induces lane
  formation
• Distance based term which stabilizes the flow
• Complexity depend on the number of grid cells

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Limitations

• Complex heterogeneous motion
• Individual path planning
• Depends on the number of groups
• Depends on the resolution
• Approximations
• Dynamic potential field
• Minimum distance enforcement to remove visual
  unpleasant artifacts

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What remains challenging

• Individual behavior planning in crowds
• Inertia walking running
• 4D eikonal equation (Position and Velocity)
• Potential function across nonuniform grids for
  speed up
• Sparse data points in areas with few people
• Finer discretizations in areas of congestion

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Summary

• Behavior planning approach to automatically
  generate realistic motions for animated
  characters
• Crowd simulation framework based on a continuum
  perspective

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Questions?
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Thank you!!