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## The Boids!¶

### Flocking¶

The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world... The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it... The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated birds.

-- Craig W. Reynolds, "Flocks, Herds, and Schools: A Distributed Behavioral Model", Computer Graphics 21 4 1987, pp 25-34 See the original paper

• Collision Avoidance: avoid collisions with nearby flockmates
• Velocity Matching: attempt to match velocity with nearby flockmates
• Flock Centering: attempt to stay close to nearby flockmates

### Setting up the Boids¶

Our boids will each have an x velocity and a y velocity, and an x position and a y position.

We'll build this up in NumPy notation, and eventually, have an animated simulation of our flying boids.

In [1]:
import numpy as np


Our positions, for each of our N boids, will be an array, shape $2 \times N$, with the x positions in the first row, and y positions in the second row.

In [2]:
boid_count = 10


We'll want to be able to seed our Boids in a random position.

We'd better define the edges of our simulation area:

In [3]:
limits = np.array([2000, 2000])

In [4]:
positions = np.random.rand(2, boid_count) * limits[:, np.newaxis]
positions

Out[4]:
array([[  167.35430282,    30.9110772 ,  1837.49053683,   793.77656223,
870.13125279,   991.17595989,  1748.36862198,   831.94148125,
408.05804795,  1746.63896938],
[  287.93987403,   414.32909831,  1855.10192652,   515.5178729 ,
1450.03841173,   318.21819801,   891.91290688,  1922.6398679 ,
291.00710946,  1096.83424046]])
In [5]:
positions.shape

Out[5]:
(2, 10)

We used broadcasting with np.newaxis to apply our upper limit to each boid. rand gives us a random number between 0 and 1. We multiply by our limits to get a number up to that limit.

In [6]:
limits[:, np.newaxis]

Out[6]:
array([[2000],
[2000]])
In [7]:
limits[:, np.newaxis].shape

Out[7]:
(2, 1)
In [8]:
np.random.rand(2, boid_count).shape

Out[8]:
(2, 10)

So we multiply a $2\times1$ array by a $2 \times 10$ array -- and get a $2\times 10$ array.

Let's put that in a function:

In [9]:
def new_flock(count, lower_limits, upper_limits):
width = upper_limits - lower_limits
return (lower_limits[:, np.newaxis] + np.random.rand(2, count) * width[:, np.newaxis])


For example, let's assume that we want our initial positions to vary between 100 and 200 in the x axis, and 900 and 1100 in the y axis. We can generate random positions within these constraints with:

positions = new_flock(boid_count, np.array([100, 900]), np.array([200, 1100]))


But each bird will also need a starting velocity. Let's make these random too:

We can reuse the new_flock function defined above, since we're again essentially just generating random numbers from given limits. This saves us some code, but keep in mind that using a function for something other than what its name indicates can become confusing!

Here, we will let the initial x velocities range over $[0, 10]$ and the y velocities over $[-20, 20]$.

In [10]:
velocities = new_flock(boid_count, np.array([0, -20]), np.array([10, 20]))
velocities

Out[10]:
array([[  0.58160645,   6.85675415,   0.79429912,   4.28217173,
6.90596489,   6.53093284,   6.1976538 ,   0.67078065,
2.81439987,   1.1232589 ],
[ 19.46204218,   1.48534274,  14.38525224,  11.3373868 ,
10.87927168,   8.41857152,   4.07100383,  -4.69135434,
-0.6210188 ,  12.63467332]])

### Flying in a Straight Line¶

Now we see the real amazingness of NumPy: if we want to move our whole flock according to

$\delta_x = \delta_t \cdot \frac{dv}{dt}$

we just do:

In [11]:
positions += velocities


### Matplotlib Animations¶

So now we can animate our Boids using the matplotlib animation tools. All we have to do is import the relevant libraries:

In [12]:
from matplotlib import animation
from matplotlib import pyplot as plt
%matplotlib inline


Then, we make a static plot, showing our first frame:

In [13]:
# create a simple plot
# initial x position in [100, 200], initial y position in [900, 1100]
# initial x velocity in [0, 10], initial y velocity in [-20, 20]
positions = new_flock(100, np.array([100, 900]), np.array([200, 1100]))
velocities = new_flock(100, np.array([0, -20]), np.array([10, 20]))

figure = plt.figure()
axes = plt.axes(xlim=(0, limits[0]), ylim=(0, limits[1]))
scatter = axes.scatter(positions[0, :], positions[1, :],
marker='o', edgecolor='k', lw=0.5)
scatter

Out[13]:
<matplotlib.collections.PathCollection at 0x2b65c578c320>

Then, we define a function which updates the figure for each timestep

In [14]:
def update_boids(positions, velocities):
positions += velocities

def animate(frame):
update_boids(positions, velocities)
scatter.set_offsets(positions.transpose())


Call FuncAnimation, and specify how many frames we want:

In [15]:
anim = animation.FuncAnimation(figure, animate,
frames=50, interval=50)


Save out the figure:

In [16]:
positions = new_flock(100, np.array([100, 900]), np.array([200, 1100]))
velocities = new_flock(100, np.array([0, -20]), np.array([10, 20]))
anim.save('boids_1.mp4')


You can even view the results directly in the notebook.

In [17]:
from IPython.display import HTML
HTML(anim.to_jshtml())

Out[17]:

Once Loop Reflect

### Fly towards the middle¶

Boids try to fly towards the middle:

In [18]:
positions = new_flock(4, np.array([100, 900]), np.array([200, 1100]))
velocities = new_flock(4, np.array([0, -20]), np.array([10, 20]))

In [19]:
positions

Out[19]:
array([[  128.07164812,   125.35480411,   147.4974637 ,   188.32382959],
[  984.4465776 ,   947.05947177,   977.81453647,  1024.51482903]])
In [20]:
velocities

Out[20]:
array([[  3.55044054,   9.52486896,   7.98208562,   9.64996727],
[ -4.30090511, -19.29949139, -14.78339043,  17.71928863]])
In [21]:
middle = np.mean(positions, 1)
middle

Out[21]:
array([ 147.31193638,  983.45885372])
In [22]:
direction_to_middle = positions - middle[:, np.newaxis]
direction_to_middle

Out[22]:
array([[-19.24028826, -21.95713227,   0.18552732,  41.01189321],
[  0.98772388, -36.39938195,  -5.64431725,  41.05597531]])

This is easier and faster than:

for bird in birds:
for dimension in [0, 1]:
direction_to_middle[dimension][bird] = positions[dimension][bird] - middle[dimension]

In [23]:
move_to_middle_strength = 0.01
velocities = velocities - direction_to_middle * move_to_middle_strength


Let's update our function, and animate that:

In [24]:
def update_boids(positions, velocities):
move_to_middle_strength = 0.01
middle = np.mean(positions, 1)
direction_to_middle = positions - middle[:, np.newaxis]
velocities -= direction_to_middle * move_to_middle_strength
positions += velocities

In [25]:
def animate(frame):
update_boids(positions, velocities)
scatter.set_offsets(positions.transpose())

In [26]:
anim = animation.FuncAnimation(figure, animate,
frames=50, interval=50)

In [27]:
positions = new_flock(100, np.array([100, 900]), np.array([200, 1100]))
velocities = new_flock(100, np.array([0, -20]), np.array([10, 20]))
HTML(anim.to_jshtml())

Out[27]:

Once Loop Reflect