.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/beginner/plot_brachistochrone.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_brachistochrone.py>`
        to download the full example code.

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

.. _sphx_glr_examples_beginner_plot_brachistochrone.py:


Brachistochrone
===============

Objective
---------

- Show how to solve a variational problem using opty with variable time
  interval.

Introduction
------------

A particle is to slide without friction under the constant force of gravity on
a curve from A to B in the shortest time. B must not be directly under A (else
the problem is trivial). The solution curve is called the Brachistochrone_
which is a cycloid.

.. _Brachistochrone: https://en.wikipedia.org/wiki/Brachistochrone_curve

Explanation of the Approach Taken
---------------------------------

Let the curve be called :math:`b(x)`. If the particle is at :math:`(x(t) ,
b(x(t))` then it cannot have a speed component normal to the tangent of the
curve at this point. This fact is used to set up the equations of motion.
Without loss of generality the starting point is (0, 0).

Notes
-----

The curve is parameterized by time and a nonholonomic constraint is introduced
to ensure the motion is always tangent to the curve. The nonholonomic
constraint is similar to the constraint present in the `Chaplygin Sleigh`_.

.. _Chaplygin Sleigh: https://en.wikipedia.org/wiki/Chaplygin_sleigh

**States**

- :math:`x`: x-coordinate of the particle
- :math:`y`: y-coordinate of the particle, opposite in sign to the
  gravitational force
- :math:`u_x`: x-component of the velocity of the particle
- :math:`u_y`: y-component of the velocity of the particle

**Inputs**

- :math:`\beta`: angle of the tangent of the curve with the horizontal

**Known Parameters**

- :math:`m`: mass of the particle
- :math:`g`: acceleration due to gravity
- :math:`b_1`: x coordinate of the final point
- :math:`b_2`: y coordinate of the final point

**Free Parameters**

- :math:`h`: time interval

.. GENERATED FROM PYTHON SOURCE LINES 62-73

.. code-block:: Python

    import os
    from opty.utils import MathJaxRepr
    import numpy as np
    import sympy as sm
    import sympy.physics.mechanics as me
    import matplotlib.pyplot as plt
    from opty import Problem
    from scipy.optimize import root
    from scipy.interpolate import CubicSpline
    from matplotlib.animation import FuncAnimation








.. GENERATED FROM PYTHON SOURCE LINES 74-83

Equations of Motion
-------------------

The equations of motion represent a particle :math:`P` of mass :math:`m`
moving in a uniform gravitational field. The point moves on a surface that
applies a normal force to it and the angle of the surface relative to the
horizontal is :math:`\beta(t)`. A nonholonomic constraint to ensure the
particle's velocity only has a component tangent to the surface is added to
the equations of motion making a set of differential algebraic equations.

.. GENERATED FROM PYTHON SOURCE LINES 83-120

.. code-block:: Python

    N, A = me.ReferenceFrame('N'), me.ReferenceFrame('A')
    O, P = sm.symbols('O P', cls=me.Point)
    O.set_vel(N, 0)
    t = me.dynamicsymbols._t

    m, g = sm.symbols('m g')
    x, y, ux, uy = me.dynamicsymbols('x y u_x u_y')
    beta = me.dynamicsymbols('beta')

    state_symbols = (x, y, ux, uy)

    A.orient_axis(N, beta, N.z)

    P.set_pos(O, x*N.x + y*N.y)
    P.set_vel(N, ux*N.x + uy*N.y)

    speed_constr = sm.Matrix([P.vel(N).dot(A.y)])

    kd = sm.Matrix([ux - x.diff(t), uy - y.diff(t)])

    bodies = [me.Particle('body', P, m)]
    forces = [(P, -m*g*N.y)]

    kane = me.KanesMethod(
        N,
        q_ind=[x, y],
        u_ind=[ux],
        u_dependent=[uy],
        kd_eqs=kd,
        velocity_constraints=speed_constr,
    )
    fr, frstar = kane.kanes_equations(bodies, loads=forces)

    eom = kd.col_join(fr + frstar)
    eom = eom.col_join(speed_constr)
    MathJaxRepr(eom)






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    $$\left[\begin{matrix}u_{x} - \dot{x}\\u_{y} - \dot{y}\\- \frac{g m \sin{\left(\beta \right)}}{\cos{\left(\beta \right)}} - \frac{m \sin{\left(\beta \right)} \dot{u}_{y}}{\cos{\left(\beta \right)}} - m \dot{u}_{x}\\- u_{x} \sin{\left(\beta \right)} + u_{y} \cos{\left(\beta \right)}\end{matrix}\right]$$
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 121-126

Set up the Optimization Problem and Solve It
--------------------------------------------

The goal is to minimize the time to reach the second point. So make the time
interval a variable :math:`h`.

.. GENERATED FROM PYTHON SOURCE LINES 126-141

.. code-block:: Python

    h = sm.symbols('h')
    num_nodes = 201
    t0, tf = 0*h, h*(num_nodes - 1)


    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 142-145

Set up the constraints and bounds, objective function and its gradient for
the problem. (b1, b2) is the final point and (0, 0) is the starting point is
(0/0).

.. GENERATED FROM PYTHON SOURCE LINES 145-177

.. code-block:: Python

    b1 = 10.0
    b2 = -6.0

    instance_constraint = (
        x.func(t0),
        y.func(t0),
        ux.func(t0),
        uy.func(t0),
        x.func(tf) - b1,
        y.func(tf) - b2,
    )

    bounds = {
        h: (0.0, 1.0),
        beta: (-np.pi/2 + 1.e-5, 0.0),
    }

    par_map = {m: 1.0, g: 9.81}

    prob = Problem(
        obj,
        obj_grad,
        eom,
        state_symbols,
        num_nodes,
        h,
        time_symbol=t,
        known_parameter_map=par_map,
        instance_constraints=instance_constraint,
        bounds=bounds,
    )








.. GENERATED FROM PYTHON SOURCE LINES 178-181

Solve the problem. Use the given solution if available, else pick a
reasonable initial guess and solve the problem. This problem takes a large
number of iterations to reach the optimal solution.

.. GENERATED FROM PYTHON SOURCE LINES 181-192

.. code-block:: Python

    fname = f'brachistochrone_{num_nodes}_nodes_solution.csv'
    if os.path.exists(fname):
        solution = np.loadtxt(fname)
    else:
        initial_guess = np.random.randn(prob.num_free)*0.1
        prob.add_option('max_iter', 15000)
        solution, info = prob.solve(initial_guess)
        print(info['status_msg'])

        _ = prob.plot_objective_value()








.. GENERATED FROM PYTHON SOURCE LINES 193-194

State and input trajectories:

.. GENERATED FROM PYTHON SOURCE LINES 194-196

.. code-block:: Python

    _ = prob.plot_trajectories(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_brachistochrone_001.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_brachistochrone_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 197-198

Constraint violations:

.. GENERATED FROM PYTHON SOURCE LINES 198-200

.. code-block:: Python

    _ = prob.plot_constraint_violations(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_brachistochrone_002.png
   :alt: Constraint violations
   :srcset: /examples/beginner/images/sphx_glr_plot_brachistochrone_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 201-203

Animate the Solution
--------------------

.. GENERATED FROM PYTHON SOURCE LINES 203-206

.. code-block:: Python

    fps = 35









.. GENERATED FROM PYTHON SOURCE LINES 207-209

Calculate the Brachistochrone from (0, 0) to (b1, b2). It is very sensitive
to the initial guess.

.. GENERATED FROM PYTHON SOURCE LINES 209-223

.. code-block:: Python

    def func(x0):
        """Gives the Brachistochrome equation for the starting point at (0, 0) and
        the final point at (b1, b2)"""
        r, theta = x0[0], x0[1]
        return [r*theta - r*np.sin(theta) - b1, r*(1.0 - np.cos(theta)) - b2]


    x0 = [1.0, 1.0]
    resultat = root(func, x0)

    times = np.linspace(0.0, resultat.x[1], num_nodes)
    xx = resultat.x[0]*times - resultat.x[0]*np.sin(times)
    yy = resultat.x[0]*(1.0 - np.cos(times))








.. GENERATED FROM PYTHON SOURCE LINES 224-225

Set up the plot.

.. GENERATED FROM PYTHON SOURCE LINES 225-266

.. code-block:: Python

    state_vals, input_vals, _, h_val = prob.parse_free(solution)
    t_arr = prob.time_vector(solution=solution)
    state_sol = CubicSpline(t_arr, state_vals.T)
    input_sol = CubicSpline(t_arr, input_vals.T)

    xmin = -1.0
    xmax = b1 + 1.0
    ymin = b2 - 1.0
    ymax = 1.0

    arrow_head = me.Point('arrow_head')
    arrow_head.set_pos(P, ux*N.x + uy*N.y)

    coordinates = P.pos_from(O).to_matrix(N)
    coordinates = coordinates.row_join(arrow_head.pos_from(O).to_matrix(N))

    coords_lam = sm.lambdify(list(state_symbols) + [beta] + list(par_map.keys()),
                             coordinates, cse=True)


    def init_plot():
        fig, ax = plt.subplots(figsize=(7, 7))
        ax.set_xlim(xmin, xmax)
        ax.set_ylim(ymin, ymax)
        ax.set_aspect('equal')
        ax.set_xlabel('x', fontsize=15)
        ax.set_ylabel('y', fontsize=15)

        # Plot the Brachistochrone analytical solution.
        ax.plot(xx, yy, color='black', lw=0.5)
        ax.scatter([0], [0], color='red', s=10)
        ax.scatter([b1], [b2], color='green', s=10)

        # Set up the point and the speed arrow.
        punkt = ax.scatter([], [], color='red', s=50)
        # Draw the speed vector.
        pfeil = ax.quiver([], [], [], [], color='green', scale=25, width=0.004)

        return fig, ax, punkt, pfeil









.. GENERATED FROM PYTHON SOURCE LINES 267-268

Function to update the plot for each animation frame

.. GENERATED FROM PYTHON SOURCE LINES 268-284

.. code-block:: Python

    fig, ax, punkt, pfeil = init_plot()


    def update(t):
        message = (f'Running time {t:.2f} sec \n'
                   f'The green arrow is the speed \n'
                   f'Slow motion video.')
        ax.set_title(message, fontsize=12)

        coords = coords_lam(*state_sol(t), input_sol(t), *par_map.values())
        punkt.set_offsets([coords[0, 0], coords[1, 0]])

        pfeil.set_offsets([coords[0, 0], coords[1, 0]])
        pfeil.set_UVC(coords[0, 1] - coords[0, 0], coords[1, 1] - coords[1, 0])





.. image-sg:: /examples/beginner/images/sphx_glr_plot_brachistochrone_003.png
   :alt: plot brachistochrone
   :srcset: /examples/beginner/images/sphx_glr_plot_brachistochrone_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 285-286

Create the animation.

.. GENERATED FROM PYTHON SOURCE LINES 286-294

.. code-block:: Python

    fig, ax, punkt, pfeil = init_plot()
    animation = FuncAnimation(
        fig,
        update,
        frames=np.arange(t0, (num_nodes + 1)*h_val, 1/fps),
        interval=3000/fps,
    )




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         /* set a timeout to make sure all the above elements are created before
            the object is initialized. */
         setTimeout(function() {
             animef3de2330ca24c119c6a7631f072e288 = new Animation(frames, img_id, slider_id, 85.0,
                                      loop_select_id);
         }, 0);
       })()
     </script>






.. GENERATED FROM PYTHON SOURCE LINES 295-297

A frame from the animation.
sphinx_gallery_thumbnail_number = 4

.. GENERATED FROM PYTHON SOURCE LINES 297-298

.. code-block:: Python

    plt.show()








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (1 minutes 0.578 seconds)


.. _sphx_glr_download_examples_beginner_plot_brachistochrone.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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