flocks, herds and schools modeling and analytic approaches
TRANSCRIPT
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Flocks, Herds and Schools
Modeling and Analytic Approaches
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The Problem
• To Model:– Flocks of Birds in 3-D– Herds of animals in 2-D– Schools of Fish in 3-D
• To Calculate / Analyze:– How structured dynamic aggregates form and
move, especially how a uniform heading is attained
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History of Approaches
• 1986: Craig Reynold– “Flocks, Herds and Schools: A distributed behavioral model”
• 1995: T. Viscek– Discrete Equation-based simulation model
• 1998: John Toner and Yuhai Tu:– “Flocks, Herds, and Schools: A quantitative theory of
flocking”
• 2002: Tanner and Jadbabaie– “Stable Flocking of Mobile Agents”
• 2002: Olfati-Saber
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Reynold’s Boids
• The proof is in the watching: – http://www.red3d.com/cwr/boids/applet/
• Point agents located in 3-D Euclidean space, with a variable heading vector: – A = (x1,x2,x3,h1,h2,h3)
• Two Basic components:– Individual Agent capabilities (geometric flight)– Inter-agent group behavior rules
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Reynolds Boids
• We won’t really look at the first component
• The second component is key. The three “Reynolds rules” are:– Collision Avoidance– Velocity Matching– Flock centering
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The Three Reynolds Rules
Separation
Coherence
Alignment (Velocity matching)
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Reynolds Boids
• The rules generate acceleration imperatives for the agent
• Accelerations are aggregated for each agent via weighting and priority (to reduce indecision) and then fed into a flight module
• Other goal-seeking and obstacle avoidance behaviors
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Reynolds Boids
• Important in its time: in 1986 the power of local algorithms was not understood
• Made for graphics community, but taken up enthusiastically by “complexity” and A-Life community
• Flocking is impromptu: flocks “emerge” from randomized initial configurations
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Reynolds Boids
• Between 1986 and 1998, there was a lot of interest in the model, and many variants of it:– Schooling (see video)– Ant behavior– Multiple species
• But very little theory.
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A Theory of “Flocking”
• Toner + Tu decided to use statistical physics
• A continuum model in various dimensions
• Flocks are modeled as a compressible statistical fluid (!)
• Theory is related to – Landau-Ginsburg Ferromagnetism– Navier-Stokes theory
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Flocks as Fluids
• “Birds” are particles without heading• Large, large numbers available to
meaningful statistics (i.e. the infinite limit)• Flocks are isotropic• Goal is to calculate correlations,
symmetries, and phase changes
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Flocks as Fluids• The equations of the model are:
where v is the bird velocity field and p is the bird density terms = Navier Stokes and terms = ensure movement P term = “Pressure” (collision avoidance) D terms = Diffusion (centering) f term = noise
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Flocks as Fluids• Main Results
– There is an ordered phase in which given appropriate statistics of initial condition, group velocity spread tends to 0
– Flocks support Goldstone modes (symmetry breaking)– Flocks support sound-like wave modes, density waves,
and shear waves– Much more long-range damping than a magnet in
velocity space– Much more density fluctuation than in other condensed
matter systems – Exact calculation of scaling exponents in 2-D, showing
that the “flock” is truly long-range symmetry broken– Broken symmetry corresponds to flock direction
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Flocks as Fluids
A picture in the ordered state.
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Flocks as Fluids• Problems in this approach:
– Assumptions are a bit weird (like horses as frictionless spheres)
– Numbers are way too large and densities are too high
– Statistics are too structured
– Motion doesn’t look like flocks! It looks like a fluid or a weird weak magnet.
– Equations of motion are somewhat unmotivated
– Predictions are untestable/inapplicable
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A Different Approach
• Consider N agents in the plane
• The important point is that there are control terms
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Flocks as Graphs
• Neighboring agents are influence each other by non-zero controls.
• Nearest-neighbor relations generate a graph• Let B(G) be the adjacency matrix of the
graph (i.e. 0 if not neighbor, 1 if it is)• L(G) = B(G) B(G)* is the graph Laplacian.
Eigenvalues correspond to topological properties of the graph.
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• Now, dynamics are generated by potential function V which takes on a unique minimum at desired positions
Control is determined via the equation:
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Flocks as Graphs
• Assume that the control connection graph is connected at all times. Then, Tanner et. al. proved using graph Laplacians and non-smooth analysis that:– All pairwise velocity differences converge to 0– And the system approaches a potential-
minimizing configurations
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Flocks as Graphs
• This model works at the right range of sizes
• It does not “overpredict”
• It gets geometry right
• It generates pictures that look pretty good.
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Unresolved Questions
• Obstacle Avoidance (Olfati-Saber)• Flock Emergence• Flock Disruption• General motion• Collective leader choice• Other non-potential based mechanisms