spiderman.ipynb

Kernel: Python 3

Modeling and Simulation in Python

Case study: Spider-Man

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import functions from the modsim.py module
from modsim import *

I'll start by getting the units we'll need from Pint.

m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
N = UNITS.newton
degree = UNITS.degree
radian = UNITS.radian

Spider-Man

In this case study we'll develop a model of Spider-Man swinging from a springy cable of webbing attached to the top of the Empire State Building. Initially, Spider-Man is at the top of a nearby building, as shown in this diagram.

The origin, O⃗, is at the base of the Empire State Building. The vector H⃗ represents the position where the webbing is attached to the building, relative to O⃗. The vector P⃗ is the position of Spider-Man relative to O⃗. And L⃗ is the vector from the attachment point to Spider-Man.

By following the arrows from O⃗, along H⃗, and along L⃗, we can see that

H⃗ + L⃗ = P⃗

So we can compute L⃗ like this:

L⃗ = P⃗ - H⃗

The goals of this case study are:

1. Implement a model of this scenario to predict Spider-Man's trajectory.

2. Choose the right time for Spider-Man to let go of the webbing in order to maximize the distance he travels before landing.

3. Choose the best angle for Spider-Man to jump off the building, and let go of the webbing, to maximize range.

I'll create a Params object to contain the quantities we'll need:

1. According to the Spider-Man Wiki, Spider-Man weighs 76 kg.

2. Let's assume his terminal velocity is 60 m/s.

3. The length of the web is 100 m.

4. The initial angle of the web is 45 degrees to the left of straight down.

5. The spring constant of the web is 40 N / m when the cord is stretched, and 0 when it's compressed.

Here's a Params object.

params = Params(height = 381 * m,
                g = 9.8 * m/s**2,
                mass = 75 * kg,
                area = 1 * m**2,
                rho = 1.2 * kg/m**3,
                v_term = 60 * m / s,
                length = 100 * m,
                angle = (270 - 45) * degree,
                k = 40 * N / m,
                t_0 = 0 * s,
                t_end = 30 * s)

Compute the initial position

def initial_condition(params):
    """Compute the initial position and velocity.
    
    params: Params object
    """
    height, length, angle = params.height, params.length, params.angle
    
    H⃗ = Vector(0, height)
    theta = angle.to(radian)
    x, y = pol2cart(theta, length)
    L⃗ = Vector(x, y)
    P⃗ = H⃗ + L⃗
    V⃗ = Vector(0, 0) * m/s
    
    return State(P⃗=P⃗, V⃗=V⃗)

initial_condition(params)

Now here's a version of make_system that takes a Params object as a parameter.

make_system uses the given value of v_term to compute the drag coefficient C_d.

def make_system(params):
    """Makes a System object for the given conditions.
    
    params: Params object
    
    returns: System object
    """
    init = initial_condition(params)
    
    mass, g = params.mass, params.g
    rho, area, v_term = params.rho, params.area, params.v_term
    C_d = 2 * mass * g / (rho * area * v_term**2)
    
    return System(params, init=init, C_d=C_d)

Let's make a System

system = make_system(params)

system.init

Drag and spring forces

Here's drag force, as we saw in Chapter 22.

def drag_force(V⃗, system):
    """Compute drag force.
    
    V⃗: velocity Vector
    system: `System` object
    
    returns: force Vector
    """
    rho, C_d, area = system.rho, system.C_d, system.area
    
    mag = rho * V⃗.mag**2 * C_d * area / 2
    direction = -V⃗.hat()
    f_drag = direction * mag
    return f_drag

V⃗_test = Vector(10, 10) * m/s

drag_force(V⃗_test, system)

And here's the 2-D version of spring force. We saw the 1-D version in Chapter 21.

def spring_force(L⃗, system):
    """Compute drag force.
    
    L⃗: Vector representing the webbing
    system: System object
    
    returns: force Vector
    """
    extension = L⃗.mag - system.length
    if magnitude(extension) < 0:
        mag = 0
    else:
        mag = system.k * extension
        
    direction = -L⃗.hat()
    f_spring = direction * mag
    return f_spring

L⃗_test = Vector(0, -system.length-1*m)

f_spring = spring_force(L⃗_test, system)

Here's the slope function, including acceleration due to gravity, drag, and the spring force of the webbing.

def slope_func(state, t, system):
    """Computes derivatives of the state variables.
    
    state: State (x, y, x velocity, y velocity)
    t: time
    system: System object with g, rho, C_d, area, mass
    
    returns: sequence (vx, vy, ax, ay)
    """
    P⃗, V⃗ = state
    g, mass = system.g, system.mass
    
    H⃗ = Vector(0, system.height)
    L⃗ = P⃗ - H⃗
    
    a_grav = Vector(0, -g)
    a_spring = spring_force(L⃗, system) / mass
    a_drag = drag_force(V⃗, system) / mass
    
    A⃗ = a_grav + a_drag + a_spring
    
    return V⃗, A⃗

As always, let's test the slope function with the initial conditions.

slope_func(system.init, 0, system)

And then run the simulation.

results, details = run_ode_solver(system, slope_func)
details

Visualizing the results

We can extract the x and y components as Series objects.

The simplest way to visualize the results is to plot x and y as functions of time.

def plot_position(P⃗):
    x = P⃗.extract('x')
    y = P⃗.extract('y')
    plot(x, label='x')
    plot(y, label='y')

    decorate(xlabel='Time (s)',
             ylabel='Position (m)')
    
plot_position(results.P⃗)

We can plot the velocities the same way.

def plot_velocity(V⃗):
    vx = V⃗.extract('x')
    vy = V⃗.extract('y')
    plot(vx, label='vx')
    plot(vy, label='vy')

    decorate(xlabel='Time (s)',
             ylabel='Velocity (m/s)')
    
plot_velocity(results.V⃗)

Another way to visualize the results is to plot y versus x. The result is the trajectory through the plane of motion.

def plot_trajectory(P⃗, **options):
    x = P⃗.extract('x')
    y = P⃗.extract('y')
    plot(x, y, **options)
    
    decorate(xlabel='x position (m)',
             ylabel='y position (m)')
    
plot_trajectory(results.P⃗, label='trajectory')

Letting go

Now let's find the optimal time for Spider-Man to let go. We have to run the simulation in two phases because the spring force changes abruptly when Spider-Man lets go, so we can't integrate through it.

Here are the parameters for Phase 1, running for 9 seconds.

params1 = Params(params, t_end=9*s)
system1 = make_system(params1)
results1, details1 = run_ode_solver(system1, slope_func)
plot_trajectory(results1.P⃗, label='Phase 1')

The final conditions from Phase 1 are the initial conditions for Phase 2.

t_final = get_last_label(results1) * s

Here's the position Vector.

init = results1.last_row()
init.P⃗

And the velocity Vector.

init.V⃗

Here is the System for Phase 2. We can turn off the spring force by setting k=0, so we don't have to write a new slope function.

system2 = System(system1, t_0=t_final, t_end=t_final+10*s, init=init, k=0*N/m)

Here's an event function that stops the simulation when Spider-Man reaches the ground.

def event_func(state, t, system):
    """Stops when y=0.
    
    state: State object
    t: time
    system: System object
    
    returns: height
    """
    P⃗, V⃗ = state
    return P⃗.y

Run Phase 2.

results2, details2 = run_ode_solver(system2, slope_func, events=event_func)

Plot the results.

plot_trajectory(results1.P⃗, label='Phase 1')
plot_trajectory(results2.P⃗, label='Phase 2')

Now we can gather all that into a function that takes t_release and V_0, runs both phases, and returns the results.

def run_two_phase(t_release, V⃗_0, params):
    """Run both phases.
    
    t_release: time when Spider-Man lets go of the webbing
    V_0: initial velocity
    """
    params1 = Params(params, t_end=t_release, V⃗_0=V⃗_0)
    system1 = make_system(params1)
    results1, details1 = run_ode_solver(system1, slope_func)

    t_0 = get_last_label(results1) * s
    t_end = t_0 + 10 * s
    init = results1.last_row()

    system2 = System(system1, t_0=t_0, t_end=t_end, init=init, k=0*N/m)
    results2, details2 = run_ode_solver(system2, slope_func, events=event_func)

    results = results1.combine_first(results2)
    return TimeFrame(results)

And here's a test run.

t_release = 9 * s
V⃗_0 = Vector(0, 0) * m/s

results = run_two_phase(t_release, V⃗_0, params)
plot_trajectory(results.P⃗)
x_final = results.P⃗.last_value().x

Animation

Here's a draw function we can use to animate the results.

xs = results.P⃗.extract('x')
ys = results.P⃗.extract('y')

def draw_func(state, t):
    set_xlim(xs)
    set_ylim(ys)
    x, y = state.P⃗
    plot(x, y, 'bo')
    decorate(xlabel='x position (m)',
             ylabel='y position (m)')

animate(results, draw_func)

Maximizing range

To find the best value of t_release, we need a function that takes possible values, runs the simulation, and returns the range.

def range_func(t_release, params):
    """Compute the final value of x.
    
    t_release: time to release web
    params: Params object
    """
    V_0 = Vector(0, 0) * m/s
    results = run_two_phase(t_release, V_0, params)
    x_final = results.P⃗.last_value().x
    print(t_release, x_final)
    return x_final

We can test it.

range_func(9*s, params)

And run it for a few values.

for t_release in linrange(3, 15, 3) * s:
    range_func(t_release, params)

Now we can use maximize_scalar to find the optimum.

bounds = [6, 12] * s
res = maximize_golden(range_func, bounds, params)

Finally, we can run the simulation with the optimal value.

best_time = res.x
V⃗_0 = Vector(0, 0) * m/s
results = run_two_phase(best_time, V⃗_0, params)
plot_trajectory(results.P⃗)
x_final = results.P⃗.last_value().x