1 / 22

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.

dwayne
Download Presentation

Flocks, Herds and Schools

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related