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

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

.. _sphx_glr_examples_beginner_plot_pendulum_swing_up_variable_duration.py:


Variable Duration Pendulum Swing Up
===================================

Objectives
----------

- Demonstrate how to make the simulation duration variable.
- Show how to use the NumPy backend which solves the problem without needing
  just-in-time C compilation.
- Demonstrate plotting the constraint violations as subplots.
- Show how to access the problem's attributes and methods inside the objective
  and gradient functions to help simplify constructing the objective and
  gradient values.

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

Given a simple pendulum that is driven by a torque about its joint axis, swing
the pendulum from hanging down to standing up in a minimal amount of time using
minimal input energy with a bounded torque magnitude.

Notes
-----

There is a mechanically identiclal system in the examples-gallery/beginner
folder that uses a fixed duration.

.. GENERATED FROM PYTHON SOURCE LINES 30-37

.. code-block:: Python

    import os
    import numpy as np
    import sympy as sm
    from opty import Problem
    import matplotlib.pyplot as plt
    import matplotlib.animation as animation








.. GENERATED FROM PYTHON SOURCE LINES 38-39

Start with defining the fixed duration and number of nodes.

.. GENERATED FROM PYTHON SOURCE LINES 39-42

.. code-block:: Python

    target_angle = np.pi
    num_nodes = 501








.. GENERATED FROM PYTHON SOURCE LINES 43-44

Symbolic equations of motion

.. GENERATED FROM PYTHON SOURCE LINES 44-55

.. code-block:: Python

    m, g, d, t, h = sm.symbols('m, g, d, t, h', real=True)
    theta, omega, T = sm.symbols('theta, omega, T', cls=sm.Function)

    state_symbols = (theta(t), omega(t))
    constant_symbols = (m, g, d)
    specified_symbols = (T(t),)

    eom = sm.Matrix([theta(t).diff() - omega(t),
                     m*d**2*omega(t).diff() + m*g*d*sm.sin(theta(t)) - T(t)])
    sm.pprint(eom)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ⎡                   d                  ⎤
    ⎢           -ω(t) + ──(θ(t))           ⎥
    ⎢                   dt                 ⎥
    ⎢                                      ⎥
    ⎢ 2   d                                ⎥
    ⎢d ⋅m⋅──(ω(t)) + d⋅g⋅m⋅sin(θ(t)) - T(t)⎥
    ⎣     dt                               ⎦




.. GENERATED FROM PYTHON SOURCE LINES 56-57

Specify the known system parameters.

.. GENERATED FROM PYTHON SOURCE LINES 57-64

.. code-block:: Python

    par_map = {
        m: 1.0,
        g: 9.81,
        d: 1.0,
    }









.. GENERATED FROM PYTHON SOURCE LINES 65-69

Specify the objective function and it's gradient. In this case, make the
problem instance the first argument and it is available to use inside the
these functions. This shows how to use ``.parse_free()``,
``.extract_values()``, and ``.fill_free()`` to manage the numerical vectors.

.. GENERATED FROM PYTHON SOURCE LINES 69-84

.. code-block:: Python

    def obj(prob, free):
        """Minimize the sum of the squares of the control torque."""
        _, T_vals, _, h_val = prob.parse_free(free)
        return h_val*np.sum(T_vals**2)


    def obj_grad(prob, free):
        T_vals = prob.extract_values(free, T(t))
        h_val = prob.extract_values(free, h)
        grad = np.zeros_like(free)
        prob.fill_free(grad, 2.0*h_val*T_vals, T(t))
        prob.fill_free(grad, np.sum(T_vals**2), h)
        return grad









.. GENERATED FROM PYTHON SOURCE LINES 85-87

Specify the symbolic instance constraints, i.e. initial and end conditions
using node numbers 0 to N - 1

.. GENERATED FROM PYTHON SOURCE LINES 87-92

.. code-block:: Python

    instance_constraints = (theta(0*h),
                            theta((num_nodes - 1)*h) - target_angle,
                            omega(0*h),
                            omega((num_nodes - 1)*h))








.. GENERATED FROM PYTHON SOURCE LINES 93-97

Create an optimization problem. If the backend is set to ``numpy``, no C
compiler is needed and the problem can be solved using pure Python code.
There is a large performance loss but for simple problems performance may not
be a concern.

.. GENERATED FROM PYTHON SOURCE LINES 97-104

.. code-block:: Python

    prob = Problem(obj, obj_grad, eom, state_symbols, num_nodes, h,
                   known_parameter_map=par_map,
                   instance_constraints=instance_constraints,
                   time_symbol=t,
                   bounds={T(t): (-2.0, 2.0), h: (0.0, 0.5)},
                   backend='numpy')








.. GENERATED FROM PYTHON SOURCE LINES 105-108

Use existing solution if available else pick a reasonable initial guess and
solve the problem. Use approximately zero as an initial guess to avoid
divide-by-zero, and solve the problem.

.. GENERATED FROM PYTHON SOURCE LINES 108-117

.. code-block:: Python

    fname = f'pendulum_swing_up_variable_duration_{num_nodes}_nodes_solution.csv'
    if os.path.exists(fname):
        solution = np.loadtxt(fname)
    else:
        initial_guess = np.full(prob.num_free, 1e-10)
        solution, info = prob.solve(initial_guess)
        print(info['status_msg'])
        print(info['obj_val'])








.. GENERATED FROM PYTHON SOURCE LINES 118-119

Plot the optimal state and input trajectories.

.. GENERATED FROM PYTHON SOURCE LINES 119-121

.. code-block:: Python

    _ = prob.plot_trajectories(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_variable_duration_001.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_variable_duration_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 122-123

Plot the constraint violations.

.. GENERATED FROM PYTHON SOURCE LINES 123-125

.. code-block:: Python

    _ = prob.plot_constraint_violations(solution, subplots=True)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_variable_duration_002.png
   :alt: Constraint violations
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_variable_duration_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 126-127

Animate the pendulum swing up.

.. GENERATED FROM PYTHON SOURCE LINES 127-161

.. code-block:: Python

    interval_value = solution[-1]
    time = prob.time_vector(solution=solution)
    angle = solution[:num_nodes]

    fig = plt.figure()
    ax = fig.add_subplot(111, aspect='equal', autoscale_on=False,
                         xlim=(-2, 2), ylim=(-2, 2))
    ax.grid()

    line, = ax.plot([], [], 'o-', lw=2)
    time_template = 'time = {:0.1f}s'
    time_text = ax.text(0.05, 0.9, '', transform=ax.transAxes)


    def init():
        line.set_data([], [])
        time_text.set_text('')
        return line, time_text


    def animate(i):
        x = [0, par_map[d]*np.sin(angle[i])]
        y = [0, -par_map[d]*np.cos(angle[i])]

        line.set_data(x, y)
        time_text.set_text(time_template.format(i*interval_value))
        return line, time_text


    ani = animation.FuncAnimation(fig, animate, range(0, num_nodes, 4),
                                  interval=int(interval_value*1000*4),
                                  blit=True, init_func=init)

    plt.show()



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

   **Total running time of the script:** (0 minutes 38.748 seconds)


.. _sphx_glr_download_examples_beginner_plot_pendulum_swing_up_variable_duration.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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