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

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

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

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

The Three Reynolds Rules

Separation

Coherence

Alignment (Velocity matching)

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

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

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.

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

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

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

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

Flocks as Fluids

A picture in the ordered state.

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

A Different Approach

• Consider N agents in the plane

• The important point is that there are control terms

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.

• Now, dynamics are generated by potential function V which takes on a unique minimum at desired positions

Control is determined via the equation:

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

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.

Unresolved Questions

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