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Pattern Formation

Hexagonal Pattern Formation

Hexagonal patterns can be formed in a simple way by mimicking particle interaction. A simple model of particle interaction is the Lennard-Jones potential, which we use in the following code in a slightly modified way. Instead of the big exponents (12 and 6), we use the exponents 4 and 2, which give us smaller but more manageable numbers.

The idea in the code is that every robot can use the neighbors structure to sense the distance and angle of every direct neighbor. Using the distance, we calculate the magnitude of the "virtual force" (attraction or repulsion) due to a neighbor (function lj_magnitude()). We then use the force magnitude and the angle to make an interaction vector (function lj_vector()), and proceed to sum all of these contributions together into an accumulator vector (functions lj_sum() and neighbors.reduce()). Finally, we scale the accumulator and feed it to the goto() function, which transforms a 2D vector into motion.

# We need this for 2D vectors
# Make sure you pass the correct include path to "bzzc -I <path1:path2> ..."
include "include/vec2.bzz"

# Lennard-Jones parameters
TARGET     = 283.0
EPSILON    = 150.0

# Lennard-Jones interaction magnitude
function lj_magnitude(dist, target, epsilon) {
    return -(epsilon / dist) * ((target / dist)^4 - (target / dist)^2)
}

# Neighbor data to LJ interaction vector
function lj_vector(rid, data) {
    return math.vec2.newp(lj_magnitude(data.distance, TARGET, EPSILON), data.azimuth)
}

# Accumulator of neighbor LJ interactions
function lj_sum(rid, data, accum) {
    return math.vec2.add(data, accum)
}

# Calculates and actuates the flocking interaction
function hexagon() {
    # Calculate accumulator
    var accum = neighbors.map(lj_vector).reduce(lj_sum, math.vec2.new(0.0, 0.0))
    if (neighbors.count() > 0) {
        math.vec2.scale(accum, 1.0 / neighbors.count())
    }

    # Move according to vector
    goto(accum.x, accum.y)
}

# Executed at init time
function init() {
}

# Executed every time step
function step() {
  hexagon()
}

# Execute at exit
function destroy() {
}

Square Pattern Formation

To form square lattice, we can build upon the previous example. The insight is to notice that, in a square lattice, we can color the nodes forming the lattice with two shades, e.g., red and blue, and then mimic the crystal structure of kitchen salt. In this structure, if two nodes have different colors, they stay at a distance D; if they have the same color, they stay at a distance D * sqrt(2).

With this idea in mind, the following script divides the robots in two swarms: those with an even id and those with an odd id. Then, using neighbors.kin() and neighbors.nonkin(), the robots can distinguish which distance to pick and calculate the correct interaction vector.

# We need this for 2D vectors
# Make sure you pass the correct include path to "bzzc -I <path1:path2> ..."
include "include/vec2.bzz"

# Lennard-Jones parameters
TARGET_KIN     = 283.0
EPSILON_KIN    = 150.0
TARGET_NONKIN  = 200.0
EPSILON_NONKIN = 100.0

# Lennard-Jones interaction magnitude
function lj_magnitude(dist, target, epsilon) {
    return -(epsilon / dist) * ((target / dist)^4 - (target / dist)^2)
}

# Neighbor data to LJ interaction vector
function lj_vector_kin(rid, data) {
    return math.vec2.newp(lj_magnitude(data.distance, TARGET_KIN, EPSILON_KIN), data.azimuth)
}

# Neighbor data to LJ interaction vector
function lj_vector_nonkin(rid, data) {
    return math.vec2.newp(lj_magnitude(data.distance, TARGET_NONKIN, EPSILON_NONKIN), data.azimuth)
}

# Accumulator of neighbor LJ interactions
function lj_sum(rid, data, accum) {
    return math.vec2.add(data, accum)
}

# Calculates and actuates the flocking interaction
function square() {
    # Calculate accumulator
    var accum = neighbors.kin().map(lj_vector_kin).reduce(lj_sum, math.vec2.new(0.0, 0.0))
    accum = neighbors.nonkin().map(lj_vector_nonkin).reduce(lj_sum, accum)
    
    if (neighbors.count() > 0) {
        math.vec2.scale(accum, 1.0 / neighbors.count())
    }

    # Move according to vector
    goto(accum.x, accum.y)
}

# Executed at init time
function init() {
    # Divide the swarm in two sub-swarms
    s1 = swarm.create(1)
    s1.select(id % 2 == 0)
    s2 = s1.others(2)
}

# Executed every time step
function step() {
    s1.exec(square)
    s2.exec(square)
}

# Execute at exit
function destroy() {
}