.. 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_fixed_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_fixed_duration.py>`
        to download the full example code.

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

.. _sphx_glr_examples_beginner_plot_pendulum_swing_up_fixed_duration.py:


Fixed Duration Pendulum Swing Up
================================

Objective
---------

- Show the use of opty with likely the simplest example possible.


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

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


Notes
-----

This example uses a fixed duration. There is an example which is mechanically
identical to this one, but it uses a variable time interval, to get the
pendulum up as fast as possible, given the torque limitations.

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


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








.. GENERATED FROM PYTHON SOURCE LINES 36-37

Start with defining the fixed duration and number of nodes.

.. GENERATED FROM PYTHON SOURCE LINES 37-41

.. code-block:: Python

    duration = 10.0  # seconds
    num_nodes = 501
    interval_value = duration/(num_nodes - 1)








.. GENERATED FROM PYTHON SOURCE LINES 42-43

Specify the symbolic equations of motion.

.. GENERATED FROM PYTHON SOURCE LINES 43-54

.. code-block:: Python

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

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

    eom = sm.Matrix([theta(t).diff() - omega(t),
                     I*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                ⎥
    ⎢                                   ⎥
    ⎢  d                                ⎥
    ⎢I⋅──(ω(t)) + d⋅g⋅m⋅sin(θ(t)) - T(t)⎥
    ⎣  dt                               ⎦




.. GENERATED FROM PYTHON SOURCE LINES 55-56

Specify the known system parameters.

.. GENERATED FROM PYTHON SOURCE LINES 56-63

.. code-block:: Python

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








.. GENERATED FROM PYTHON SOURCE LINES 64-66

Specify the objective function and it's gradient, in this case it calculates
the area under the input torque curve over the simulation.

.. GENERATED FROM PYTHON SOURCE LINES 66-74

.. code-block:: Python

    obj_func = sm.Integral(T(t)**2, t)
    sm.pprint(obj_func)
    obj, obj_grad = create_objective_function(obj_func, state_symbols,
                                              specified_symbols, tuple(),
                                              num_nodes,
                                              interval_value,
                                              time_symbol=t)





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

 .. code-block:: none

    ⌠         
    ⎮  2      
    ⎮ T (t) dt
    ⌡         




.. GENERATED FROM PYTHON SOURCE LINES 75-78

Specify the symbolic instance constraints, i.e. initial and end conditions,
where the pendulum starts a zero degrees (hanging down) and ends at 180
degrees (standing up).

.. GENERATED FROM PYTHON SOURCE LINES 78-86

.. code-block:: Python

    target_angle = np.pi  # radians
    instance_constraints = (
        theta(0.0),
        theta(duration) - target_angle,
        omega(0.0),
        omega(duration),
    )








.. GENERATED FROM PYTHON SOURCE LINES 87-88

Limit the torque to a maximum magnitude.

.. GENERATED FROM PYTHON SOURCE LINES 88-90

.. code-block:: Python

    bounds = {T(t): (-2.0, 2.0)}








.. GENERATED FROM PYTHON SOURCE LINES 91-92

Create an optimization problem.

.. GENERATED FROM PYTHON SOURCE LINES 92-102

.. code-block:: Python

    prob = Problem(obj, obj_grad, eom, state_symbols, num_nodes, interval_value,
                   known_parameter_map=par_map,
                   instance_constraints=instance_constraints,
                   bounds=bounds,
                   time_symbol=t,
                   backend='numpy')

    # Use a random positive initial guess.
    initial_guess = np.random.randn(prob.num_free)








.. GENERATED FROM PYTHON SOURCE LINES 103-104

Find the optimal solution.

.. GENERATED FROM PYTHON SOURCE LINES 104-108

.. code-block:: Python

    solution, info = prob.solve(initial_guess)
    print(info['status_msg'])
    print(info['obj_val'])





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

 .. code-block:: none

    b'Algorithm terminated successfully at a locally optimal point, satisfying the convergence tolerances (can be specified by options).'
    15.641553623553241




.. GENERATED FROM PYTHON SOURCE LINES 109-110

Plot the sparsity pattern of the Jacobian.

.. GENERATED FROM PYTHON SOURCE LINES 110-112

.. code-block:: Python

    _ = prob.plot_jacobian_sparsity()




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_001.png
   :alt: plot pendulum swing up fixed duration
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 113-114

Plot the optimal state and input trajectories.

.. GENERATED FROM PYTHON SOURCE LINES 114-116

.. code-block:: Python

    _ = prob.plot_trajectories(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_002.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 117-118

Plot the constraint violations.

.. GENERATED FROM PYTHON SOURCE LINES 118-120

.. code-block:: Python

    _ = prob.plot_constraint_violations(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_003.png
   :alt: Constraint violations
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 121-122

Plot the objective function as a function of optimizer iteration.

.. GENERATED FROM PYTHON SOURCE LINES 122-124

.. code-block:: Python

    _ = prob.plot_objective_value()




.. image-sg:: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_004.png
   :alt: Objective Value
   :srcset: /examples/beginner/images/sphx_glr_plot_pendulum_swing_up_fixed_duration_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 125-126

Animate the pendulum swing up.

.. GENERATED FROM PYTHON SOURCE LINES 126-159

.. code-block:: Python

    time = prob.time_vector()
    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(time[i]))
        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 42.985 seconds)


.. _sphx_glr_download_examples_beginner_plot_pendulum_swing_up_fixed_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_fixed_duration.ipynb <plot_pendulum_swing_up_fixed_duration.ipynb>`

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

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

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

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


.. only:: html

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

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