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Flocks, Herds and Schools

Flocks, Herds and Schools. Modeling and Analytic Approaches. 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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Flocks, Herds and Schools

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  1. Flocks, Herds and Schools Modeling and Analytic Approaches

  2. 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

  3. 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

  4. 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

  5. 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

  6. Separation The Three Reynolds Rules Coherence Alignment (Velocity matching)

  7. 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

  8. 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

  9. 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.

  10. 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

  11. 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

  12. 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

  13. 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

  14. Flocks as Fluids A picture in the ordered state.

  15. 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

  16. A Different Approach • Consider N agents in the plane • The important point is that there are control terms

  17. 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.

  18. Now, dynamics are generated by potential function V which takes on a unique minimum at desired positions Control is determined via the equation:

  19. 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

  20. 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.

  21. Unresolved Questions • Obstacle Avoidance (Olfati-Saber) • Flock Emergence • Flock Disruption • General motion • Collective leader choice • Other non-potential based mechanisms

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