.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/beginner/plot_hilly_race.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_beginner_plot_hilly_race.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_beginner_plot_hilly_race.py:


Hilly Race
==========

Simulation of a particle subject to the force of gravity and air drag as it
traverses an elevation profile with a specified numerical shape to reach the
course finish line in minimal time.

Objectives
----------

- Demonstrate generating a known trajectory via numerical functions.
- Show how interpolation can be used to generated a specified input.

.. GENERATED FROM PYTHON SOURCE LINES 16-24

.. code-block:: Python

    import numpy as np
    import sympy as sm
    import sympy.physics.mechanics as me
    import matplotlib.pyplot as plt
    import matplotlib.animation as animation
    from opty import Problem
    from opty.utils import MathJaxRepr








.. GENERATED FROM PYTHON SOURCE LINES 25-42

Define the variables and equations of motion.

- :math:`m`: particle mass [kg]
- :math:`g`: acceleration due to gravity [m/s/s]
- :math:`h`: time step [s]
- :math:`s(t)`: distance traveled along elevation profile [m]
- :math:`v(t)`: longitudinal speed [m/s]
- :math:`x(t)`: horizontal coordinate [m]
- :math:`y(t)`: vertical coordinate [m]
- :math:`p(t)`: propulsion power [W]
- :math:`e(t)`: propulsion energy [J]
- :math:`\theta(x(t))`: slope angle [rad]

Note that the slope angle :math:`\theta` is made a function of :math:`x(t)`
instead of simply :math:`t`. Numerical functions that generate
:math:`\theta(x(t))` and :math:`\frac{d\theta}{dx}` will be supplied to
incorporate this known trajectory into the equations of motion.

.. GENERATED FROM PYTHON SOURCE LINES 42-57

.. code-block:: Python

    m, g, h = sm.symbols('m, g, h', real=True)
    s, v, x, y, p, e = me.dynamicsymbols('s, v, x, y, p, e', real=True)
    theta = sm.Function('theta')(x)

    states = (x, y, s, v, e)

    eom = sm.Matrix([
        x.diff() - v*sm.cos(theta),
        y.diff() - v*sm.sin(theta),
        s.diff() - v,
        m*v.diff() - p/v + m*g*sm.sin(theta) + v**2/3,
        e.diff() - p,
    ])
    MathJaxRepr(eom)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    $$\left[\begin{matrix}- v \cos{\left(\theta{\left(x \right)} \right)} + \dot{x}\\- v \sin{\left(\theta{\left(x \right)} \right)} + \dot{y}\\- v + \dot{s}\\g m \sin{\left(\theta{\left(x \right)} \right)} + m \dot{v} - \frac{p}{v} + \frac{v^{2}}{3}\\- p + \dot{e}\end{matrix}\right]$$
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 58-68

In the equations of motion, :math:`\theta(x(t))` is present and will be
specified as a known trajectory. When the Jacobian of the NLP constraint
function is generated, :math:`\frac{d \theta}{dx}` will also be required, so
it will also be specified as a known trajectory.

The elevation profile of a race course is often derived from measurements of
a road surface. If a series of elevation values at specified linear distances
are available, the slope is then also a function of the linear distances. The
following code creates an elevation profile that simulates data collected
from a road surface, albeit artificially smooth.

.. GENERATED FROM PYTHON SOURCE LINES 68-85

.. code-block:: Python

    amp = 50.0
    omega = 2*np.pi/500.0  # one period every 500 meters
    x_meas = np.linspace(-250.0, 1250.0, num=3001)  # extend beyond expected range
    y_meas = amp*np.sin(omega*x_meas)
    theta_meas = np.arctan(amp*omega*np.cos(omega*x_meas))
    dthetadx_meas = -amp*omega**2*np.sin(omega*x_meas)/(
        amp**2*omega**2*np.cos(omega*x_meas)**2 + 1)

    fig, axes = plt.subplots(3, sharex=True)
    axes[0].plot(x_meas, y_meas)
    axes[0].set_ylabel(r'$y$ [m]')
    axes[1].plot(x_meas, np.rad2deg(theta_meas))
    axes[1].set_ylabel(r'$\theta$ [deg]')
    axes[2].plot(x_meas, np.rad2deg(dthetadx_meas))
    axes[2].set_ylabel(r'$\frac{d\theta}{dx}$ [deg/m]')
    _ = axes[2].set_xlabel(r'$x$ [m]')




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_001.png
   :alt: plot hilly race
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 86-90

The following functions output the slope and its derivative with respect to
:math:`x(t)` at all values of :math:`x(t)` using interpolation. The only
input to these functions should be the optimization free vector and the output
should be an array of values for :math:`\theta(t)`, one value for each node.

.. GENERATED FROM PYTHON SOURCE LINES 90-123

.. code-block:: Python

    N = 201


    def calc_theta(free):
        """
        Parameters
        ==========
        free : ndarray, shape(nN + qN + r + s, )

        Returns
        =======
        theta : ndarray, shape(N, )

        """
        x = free[0:N]
        return np.interp(x, x_meas, theta_meas)


    def calc_dthetadx(free):
        """
        Parameters
        ==========
        free : ndarray, shape(nN + qN + r + s, )

        Returns
        =======
        dthetadx : ndarray, shape(N, )

        """
        x = free[0:N]
        return np.interp(x, x_meas, dthetadx_meas)









.. GENERATED FROM PYTHON SOURCE LINES 124-126

Minimize the time to reach the final distance traveled :math:`s(t_f)` when
starting from a standstill. The time step is the last value in free.

.. GENERATED FROM PYTHON SOURCE LINES 126-136

.. code-block:: Python

    def obj(free):
        return free[-1]


    def obj_grad(free):
        grad = np.zeros_like(free)
        grad[-1] = 1.0
        return grad









.. GENERATED FROM PYTHON SOURCE LINES 137-138

Start from a zero motion state and set the race duration to 1 kilometer.

.. GENERATED FROM PYTHON SOURCE LINES 138-151

.. code-block:: Python

    t0, tf = 0*h, (N - 1)*h
    sf = 1000.0  # meters
    ef = 120000.0  # joules

    instance_constraint = (
        x.func(t0),
        y.func(t0),
        s.func(t0),
        v.func(t0),
        e.func(t0),
        s.func(tf) - sf,
    )








.. GENERATED FROM PYTHON SOURCE LINES 152-153

Limit the power and energy and make sure the speed and time step is positive.

.. GENERATED FROM PYTHON SOURCE LINES 153-160

.. code-block:: Python

    bounds = {
        h: (0.0, 10.0),
        p: (0.0, 1000.0),
        v: (0.0, np.inf),
        e: (0.0, ef),
    }








.. GENERATED FROM PYTHON SOURCE LINES 161-165

The slope angle :math:`\theta` and its derivative :math:`\frac{d\theta}{dx}`
are set as a known trajectories and the functions ``calc_theta`` and
``calc_dthetadx`` functions are provided to generate the array of values
dynamically during the optimization iterations.

.. GENERATED FROM PYTHON SOURCE LINES 165-187

.. code-block:: Python

    prob = Problem(
        obj,
        obj_grad,
        eom,
        states,
        N,
        h,
        known_parameter_map={
            m: 100.0,  # kg
            g: 9.81,  # m/s/s
        },
        known_trajectory_map={
            theta.diff(x): calc_dthetadx,  # rad/m
            theta: calc_theta,  # rad
        },
        time_symbol=me.dynamicsymbols._t,
        instance_constraints=instance_constraint,
        bounds=bounds,
        integration_method='midpoint',
        backend='numpy',
    )








.. GENERATED FROM PYTHON SOURCE LINES 188-189

Provide a linear initial guesses for each variable.

.. GENERATED FROM PYTHON SOURCE LINES 189-200

.. code-block:: Python

    initial_guess = np.random.random(prob.num_free)
    prob.fill_free(initial_guess, np.linspace(0.0, sf, num=N), x)
    prob.fill_free(initial_guess, np.zeros(N), y)
    prob.fill_free(initial_guess, np.linspace(0.0, sf, num=N), s)
    prob.fill_free(initial_guess, 10.0*np.ones(N), v)
    prob.fill_free(initial_guess, np.linspace(0.0, ef, num=N), e)
    prob.fill_free(initial_guess, 500.0*np.ones(N), p)
    prob.fill_free(initial_guess, 0.1, h)

    _ = prob.plot_trajectories(initial_guess)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_002.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 201-202

Solve the problem.

.. GENERATED FROM PYTHON SOURCE LINES 202-206

.. code-block:: Python

    solution, info = prob.solve(initial_guess)

    _ = prob.plot_objective_value()




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_003.png
   :alt: Objective Value
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 207-208

Plot the constraint violations.

.. GENERATED FROM PYTHON SOURCE LINES 208-210

.. code-block:: Python

    _ = prob.plot_constraint_violations(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_004.png
   :alt: Constraint violations
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 211-212

Plot the state and input solution trajectories.

.. GENERATED FROM PYTHON SOURCE LINES 212-214

.. code-block:: Python

    _ = prob.plot_trajectories(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_005.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_005.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 215-216

Plot the slope angle in degrees versus the horizontal distance.

.. GENERATED FROM PYTHON SOURCE LINES 216-223

.. code-block:: Python

    x_vals = prob.extract_values(solution, x)
    theta_vals = calc_theta(solution)
    fig, ax = plt.subplots()
    ax.plot(x_vals, np.rad2deg(theta_vals))
    ax.set_xlabel(x)
    _ = ax.set_ylabel(theta)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_hilly_race_006.png
   :alt: plot hilly race
   :srcset: /examples/beginner/images/sphx_glr_plot_hilly_race_006.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 224-226

Animation of the particle traversing the profile:
sphinx_gallery_thumbnail_number = 7

.. GENERATED FROM PYTHON SOURCE LINES 226-244

.. code-block:: Python

    xs, rs, ps, dh = prob.parse_free(solution)

    fig, ax = plt.subplots()
    ax.plot(xs[0], xs[1])
    dot, = ax.plot(xs[0, 0], xs[1, 0], marker='o', markersize=10)
    ax.set_aspect('equal')
    ax.set_ylabel(r'$y$ [m]')
    ax.set_xlabel(r'$x$ [m]')


    def animate(i):
        xi, yi = xs[0, i], xs[1, i]
        dot.set_data([xi], [yi])


    ani = animation.FuncAnimation(fig, animate, range(0, N, 4))

    plt.show()



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         /* set a timeout to make sure all the above elements are created before
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.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 23.796 seconds)


.. _sphx_glr_download_examples_beginner_plot_hilly_race.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_hilly_race.ipynb <plot_hilly_race.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_hilly_race.py <plot_hilly_race.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_hilly_race.zip <plot_hilly_race.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_