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

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

.. _sphx_glr_examples_beginner_plot_light_diffraction.py:


Light Diffraction
=================

Objectives
----------

- Show how to use inequalities on the equations of motion.
- Show how to use differentiable functions to approximate their non-
  differentiable counterparts.
- Show that values of the parameter map may be changed in a loop without having
  to build ``Problem`` again.

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

A particle should move from the origin to the final point in minimum time.
The maximum speed is limited to different values in different regions.
This is accomplished by using a smooth approximation of a 'hump' function in
connection with inequality constraints on the equations of motion.

Notes
-----

- The bending of light on surfaces is explained by the assumption that its
  speed is different in different media, while it 'tries' to minimize the time
  it takes to travel. More details may be found here.

  https://en.wikipedia.org/wiki/Refractive_index
  So, this may be considered as a simulation of light diffraction.

- Here differentiable hump functions are used instead of their non-
  differentiable counterparts. This may create unwanted 'edge effects' in the
  solution of the problem. Here :math:`u_x > 0.0` is enforced to avoid vertical
  motion.


**states**

- :math:`x, y` : position of the particle
- :math:`u_x, u_y` : velocity of the particle


**controls**

- :math:`f_x, f_y` : force applied to the particle

**parameters**

- :math:`m` : mass of the particle [kg]
- :math:`a_1, a_2, a_3` : boundaries of the regions [m]
- :math:`\mu_1, \mu_2, \mu_3` : maximum speed in the regions [m/s]
- :math:`\textrm{steepness}` : steepness of the 'hump' function

.. GENERATED FROM PYTHON SOURCE LINES 58-69

.. code-block:: Python


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








.. GENERATED FROM PYTHON SOURCE LINES 70-71

Define the smooth hump function.

.. GENERATED FROM PYTHON SOURCE LINES 71-79

.. code-block:: Python



    def smooth_hump(x, a, b, k):
        """gives approx 1.0 for a < x < b, 0.0 otherwise.
        The larger k, the steeper the transition"""
        return 0.5 * (sm.tanh(k * (x - a)) - sm.tanh(k * (x - b)))









.. GENERATED FROM PYTHON SOURCE LINES 80-82

Set Up the Equations of Motion, Kane's Method
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 82-106

.. code-block:: Python

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

    x, y, ux, uy = me.dynamicsymbols('x y u_x u_y')
    fx, fy = me.dynamicsymbols('f_x f_y')

    m, a1, a2, a3 = sm.symbols('m a_1 a_2 a_3')
    mu1, mu2, mu3 = sm.symbols('mu_1 mu_2 mu_3')
    steepness = sm.symbols('steepness')
    friction = sm.symbols('friction')

    P.set_pos(O, x * N.x + y * N.y)
    P.set_vel(N, ux * N.x + uy * N.y)
    bodies = [me.Particle('P', P, m)]

    forces = [(P, fx * N.x + fy * N.y - ux * friction * N.x - uy * friction * N.y)]

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

    KM = me.KanesMethod(N, q_ind=[x, y], u_ind=[ux, uy], kd_eqs=kd)
    fr, frstar = KM.kanes_equations(bodies, forces)
    eom = kd.col_join(fr + frstar)







.. GENERATED FROM PYTHON SOURCE LINES 107-108

Add the eoms to bound the speeds in the different regions.

.. GENERATED FROM PYTHON SOURCE LINES 108-118

.. code-block:: Python

    hump1 = smooth_hump(x, -1.0, a1, steepness)
    hump2 = smooth_hump(x, a1, a2, steepness)
    hump3 = smooth_hump(x, a2, a3, steepness)
    speed_mag = sm.sqrt(ux**2 + uy**2)

    eom1 = sm.Matrix([speed_mag * hump1,
                      speed_mag * hump2,
                      speed_mag * hump3])
    eom = eom.col_join(eom1)








.. GENERATED FROM PYTHON SOURCE LINES 119-120

Bound the magnitude of the driving force, but not its direction.

.. GENERATED FROM PYTHON SOURCE LINES 120-124

.. code-block:: Python

    eom = eom.col_join(sm.Matrix([sm.sqrt(fx**2 + fy**2)]))
    print((f'The equations of motion have {sm.count_ops(eom)} operations and have'
          f' shape {eom.shape}'))
    MathJaxRepr(eom)




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

 .. code-block:: none

    The equations of motion have 88 operations and have shape (8, 1)


.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    $$\left[\begin{matrix}u_{x} - \dot{x}\\u_{y} - \dot{y}\\- friction u_{x} - m \dot{u}_{x} + f_{x}\\- friction u_{y} - m \dot{u}_{y} + f_{y}\\\sqrt{u_{x}^{2} + u_{y}^{2}} \left(- 0.5 \tanh{\left(steepness \left(- a_{1} + x\right) \right)} + 0.5 \tanh{\left(steepness \left(x + 1.0\right) \right)}\right)\\\sqrt{u_{x}^{2} + u_{y}^{2}} \left(0.5 \tanh{\left(steepness \left(- a_{1} + x\right) \right)} - 0.5 \tanh{\left(steepness \left(- a_{2} + x\right) \right)}\right)\\\sqrt{u_{x}^{2} + u_{y}^{2}} \left(0.5 \tanh{\left(steepness \left(- a_{2} + x\right) \right)} - 0.5 \tanh{\left(steepness \left(- a_{3} + x\right) \right)}\right)\\\sqrt{f_{x}^{2} + f_{y}^{2}}\end{matrix}\right]$$
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 125-127

Set Up the Optimization Problem
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 127-146

.. code-block:: Python

    h = sm.symbols('h')
    interval = h
    num_nodes = 501
    t0, tf = 0.0, h * (num_nodes - 1)

    state_symbols = [x, y, ux, uy]

    par_map = {
        m: 1.0,
        a1: 3.0,
        a2: 7.0,
        a3: 11.0,
        mu1: 5.0,
        mu2: 1.0,
        mu3: 5.0,
        steepness: 80.0,
        friction: 1.0,
    }








.. GENERATED FROM PYTHON SOURCE LINES 147-148

Plot the hump function.

.. GENERATED FROM PYTHON SOURCE LINES 148-160

.. code-block:: Python

    a, b, xt, k = sm.symbols('a b xt k')
    eval_hump = sm.lambdify((xt, a, b, k), smooth_hump(xt, a, b, k))
    x_vals = np.linspace(-1, 1, 100)
    k = par_map[steepness]
    fig, ax = plt.subplots(figsize=(8, 1.5), layout='constrained')
    ax.plot(x_vals, eval_hump(x_vals, -0.5, 0.75, k))
    ax.axvline(-0.5, color='black', lw=0.5)
    ax.axvline(0.75, color='black', lw=0.5)
    ax.axhline(0.0, color='black', lw=0.5)
    ax.axhline(1.0, color='black', lw=0.5)
    _ = ax.set_title(f"Smooth hump function with steepness = {k}")




.. image-sg:: /examples/beginner/images/sphx_glr_plot_light_diffraction_001.png
   :alt: Smooth hump function with steepness = 80.0
   :srcset: /examples/beginner/images/sphx_glr_plot_light_diffraction_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 161-162

Build ``Problem``.

.. GENERATED FROM PYTHON SOURCE LINES 162-215

.. code-block:: Python



    def obj(free):
        """minimize the variable time interval."""
        return free[-1]


    def obj_grad(free):
        """Gradient of the objective function."""
        grad = np.zeros_like(free)
        grad[-1] = 1.0
        return grad


    instance_constraints = (
        x.func(t0) - 0.0,
        y.func(t0) - 0.0,
        x.func(tf) - 10.0,
        y.func(tf) - 10.0,
    )

    bounds = {
        h: (0.0, 1.0),
        x: (0.0, 10.0),
        y: (0.0, 10.0),
        ux: (0.5, np.inf),
        uy: (0.1, np.inf),
    }

    limit = 400
    eom_bounds = {
        4: (0.0, par_map[mu1]),
        5: (0.0, par_map[mu2]),
        6: (0.0, par_map[mu3]),
        7: (0.0, limit),
    }

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

    prob.add_option('max_iter', 15000)








.. GENERATED FROM PYTHON SOURCE LINES 216-217

Use existing solution if available, else solve the problem.

.. GENERATED FROM PYTHON SOURCE LINES 217-245

.. code-block:: Python

    fname = f'light_diffraction_{num_nodes}_nodes_solution.csv'

    if os.path.exists(fname):
        # Use existing solution.
        solution = np.loadtxt(fname)
    else:
        # Solve the problem. Pick a reasonable initial guess.
        initial_guess = np.ones(prob.num_free) * 0.5
        x_values = np.linspace(0, 10, num_nodes)
        y_values = np.linspace(0, 10, num_nodes)
        initial_guess[:num_nodes] = x_values
        initial_guess[num_nodes:2*num_nodes] = y_values

        it0 = 0
        for i in range(15):
            # Values of par_map may be changed in a loop without having to build
            # Problem again.
            par_map[steepness] = 10 + 5 * i

            solution, info = prob.solve(initial_guess)
            initial_guess = solution
            print(info['status_msg'])
            it1 = len(prob.obj_value)
            print('Iterations needed', it1 - it0)
            it0 = it1
        _ = prob.plot_objective_value()
        print(f'Total iterations needed were: {len(prob.obj_value)}')








.. GENERATED FROM PYTHON SOURCE LINES 246-247

Plot trajectories.

.. GENERATED FROM PYTHON SOURCE LINES 247-249

.. code-block:: Python

    _ = prob.plot_trajectories(solution, show_bounds=True)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_light_diffraction_002.png
   :alt: State Trajectories, Input Trajectories
   :srcset: /examples/beginner/images/sphx_glr_plot_light_diffraction_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 250-251

PLot the constraint violations.

.. GENERATED FROM PYTHON SOURCE LINES 251-253

.. code-block:: Python

    _ = prob.plot_constraint_violations(solution)




.. image-sg:: /examples/beginner/images/sphx_glr_plot_light_diffraction_003.png
   :alt: Constraint violations
   :srcset: /examples/beginner/images/sphx_glr_plot_light_diffraction_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 254-256

Animate the Motion
------------------

.. GENERATED FROM PYTHON SOURCE LINES 256-337

.. code-block:: Python

    fps = 15

    state_vals, input_vals, _, h_val = prob.parse_free(solution)
    tf = h_val * (num_nodes - 1)
    t_arr = np.linspace(t0, tf, num_nodes)
    state_sol = CubicSpline(t_arr, state_vals.T)
    input_sol = CubicSpline(t_arr, input_vals.T)

    # create additional point for the speed vector
    arrow_head = sm.symbols('arrow_head', cls=me.Point)
    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))

    pL, pL_vals = zip(*par_map.items())
    coords_lam = sm.lambdify((*state_symbols, fx, fy, *pL), coordinates,
                             cse=True)


    def init():
        # needed to give the picture the right size.
        xmin, xmax = -1.0, 11.
        ymin, ymax = -1.0, 11.

        fig = plt.figure(figsize=(8, 8))
        ax = fig.add_subplot(111)
        ax.set_xlim(xmin, xmax)
        ax.set_ylim(ymin, ymax)
        ax.set_aspect('equal')
        ax.grid()

        ax.axvline(xmin, color='black', lw=1)
        ax.axvline(par_map[a1], color='black', lw=1)
        ax.axvline(par_map[a2], color='black', lw=1)
        ax.axvline(par_map[a3], color='black', lw=1)
        ax.scatter(0.0, 0.0, color='red', s=20)
        ax.scatter(10.0, 10.0, color='green', s=20)

        yy = np.linspace(ymin, ymax, 100)
        ax.fill_betweenx(yy, xmin, par_map[a1], color='gray',
                         alpha=1.0/par_map[mu1])
        ax.fill_betweenx(yy, par_map[a1], par_map[a2], color='gray',
                         alpha=1.0/par_map[mu2])
        ax.fill_betweenx(yy, par_map[a2], par_map[a3], color='gray',
                         alpha=1.0/par_map[mu3])

        # Initialize the block
        line1 = ax.scatter([], [], color='blue', s=100)
        line2, = ax.plot([], [], color='red', lw=0.5)
        pfeil = ax.quiver([], [], [], [], color='green', scale=25,
                          width=0.004, headwidth=8)

        return fig, ax, line1, line2, pfeil


    fig, ax, line1, line2, pfeil = init()


    def update(t):
        message = (f'running time {t:.2f} sec \n Darker shade is slower maximum'
                   f' speed \n The speed is the green arrow')
        ax.set_title(message, fontsize=12)

        coords = coords_lam(*state_sol(t), *input_sol(t), *pL_vals)

        koords_x = []
        koords_y = []
        for t1 in np.linspace(t0, tf, int(fps * (tf - t0))):
            if t1 <= t:
                coords = coords_lam(*state_sol(t1), *input_sol(t1), *pL_vals)
                koords_x.append(coords[0, 0])
                koords_y.append(coords[1, 0])
        line2.set_data(koords_x, koords_y)
        line1.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])


    frames = np.linspace(t0, tf, int(fps * (tf - t0)))
    animation = FuncAnimation(fig, update, frames=frames, interval=1000 / fps)



.. container:: sphx-glr-animation

    .. raw:: html

        
     <link rel="stylesheet"
     href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
     <script language="javascript">
       function isInternetExplorer() {
         ua = navigator.userAgent;
         /* MSIE used to detect old browsers and Trident used to newer ones*/
         return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
       }

       /* Define the Animation class */
       function Animation(frames, img_id, slider_id, interval, loop_select_id){
         this.img_id = img_id;
         this.slider_id = slider_id;
         this.loop_select_id = loop_select_id;
         this.interval = interval;
         this.current_frame = 0;
         this.direction = 0;
         this.timer = null;
         this.frames = new Array(frames.length);

         for (var i=0; i<frames.length; i++)
         {
          this.frames[i] = new Image();
          this.frames[i].src = frames[i];
         }
         var slider = document.getElementById(this.slider_id);
         slider.max = this.frames.length - 1;
         if (isInternetExplorer()) {
             // switch from oninput to onchange because IE <= 11 does not conform
             // with W3C specification. It ignores oninput and onchange behaves
             // like oninput. In contrast, Microsoft Edge behaves correctly.
             slider.setAttribute('onchange', slider.getAttribute('oninput'));
             slider.setAttribute('oninput', null);
         }
         this.set_frame(this.current_frame);
       }

       Animation.prototype.get_loop_state = function(){
         var button_group = document[this.loop_select_id].state;
         for (var i = 0; i < button_group.length; i++) {
             var button = button_group[i];
             if (button.checked) {
                 return button.value;
             }
         }
         return undefined;
       }

       Animation.prototype.set_frame = function(frame){
         this.current_frame = frame;
         document.getElementById(this.img_id).src =
                 this.frames[this.current_frame].src;
         document.getElementById(this.slider_id).value = this.current_frame;
       }

       Animation.prototype.next_frame = function()
       {
         this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
       }

       Animation.prototype.previous_frame = function()
       {
         this.set_frame(Math.max(0, this.current_frame - 1));
       }

       Animation.prototype.first_frame = function()
       {
         this.set_frame(0);
       }

       Animation.prototype.last_frame = function()
       {
         this.set_frame(this.frames.length - 1);
       }

       Animation.prototype.slower = function()
       {
         this.interval /= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
       }

       Animation.prototype.faster = function()
       {
         this.interval *= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
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       Animation.prototype.anim_step_forward = function()
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           if(loop_state == "loop"){
             this.first_frame();
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             this.last_frame();
             this.reverse_animation();
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             this.pause_animation();
             this.last_frame();
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       Animation.prototype.anim_step_reverse = function()
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         this.current_frame -= 1;
         if(this.current_frame >= 0){
           this.set_frame(this.current_frame);
         }else{
           var loop_state = this.get_loop_state();
           if(loop_state == "loop"){
             this.last_frame();
           }else if(loop_state == "reflect"){
             this.first_frame();
             this.play_animation();
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             this.pause_animation();
             this.first_frame();
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       }

       Animation.prototype.pause_animation = function()
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         this.direction = 0;
         if (this.timer){
           clearInterval(this.timer);
           this.timer = null;
         }
       }

       Animation.prototype.play_animation = function()
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         this.pause_animation();
         this.direction = 1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_forward();
         }, this.interval);
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       Animation.prototype.reverse_animation = function()
       {
         this.pause_animation();
         this.direction = -1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_reverse();
         }, this.interval);
       }
     </script>

     <style>
     .animation {
         display: inline-block;
         text-align: center;
     }
     input[type=range].anim-slider {
         width: 374px;
         margin-left: auto;
         margin-right: auto;
     }
     .anim-buttons {
         margin: 8px 0px;
     }
     .anim-buttons button {
         padding: 0;
         width: 36px;
     }
     .anim-state label {
         margin-right: 8px;
     }
     .anim-state input {
         margin: 0;
         vertical-align: middle;
     }
     </style>

     <div class="animation">
       <img id="_anim_img31395ce0cf154094ad2a067793970eba">
       <div class="anim-controls">
         <input id="_anim_slider31395ce0cf154094ad2a067793970eba" type="range" class="anim-slider"
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         <div class="anim-buttons">
           <button title="Decrease speed" aria-label="Decrease speed" onclick="anim31395ce0cf154094ad2a067793970eba.slower()">
               <i class="fa fa-minus"></i></button>
           <button title="First frame" aria-label="First frame" onclick="anim31395ce0cf154094ad2a067793970eba.first_frame()">
             <i class="fa fa-fast-backward"></i></button>
           <button title="Previous frame" aria-label="Previous frame" onclick="anim31395ce0cf154094ad2a067793970eba.previous_frame()">
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               <i class="fa fa-play fa-flip-horizontal"></i></button>
           <button title="Pause" aria-label="Pause" onclick="anim31395ce0cf154094ad2a067793970eba.pause_animation()">
               <i class="fa fa-pause"></i></button>
           <button title="Play" aria-label="Play" onclick="anim31395ce0cf154094ad2a067793970eba.play_animation()">
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           <button title="Next frame" aria-label="Next frame" onclick="anim31395ce0cf154094ad2a067793970eba.next_frame()">
               <i class="fa fa-step-forward"></i></button>
           <button title="Last frame" aria-label="Last frame" onclick="anim31395ce0cf154094ad2a067793970eba.last_frame()">
               <i class="fa fa-fast-forward"></i></button>
           <button title="Increase speed" aria-label="Increase speed" onclick="anim31395ce0cf154094ad2a067793970eba.faster()">
               <i class="fa fa-plus"></i></button>
         </div>
         <form title="Repetition mode" aria-label="Repetition mode" action="#n" name="_anim_loop_select31395ce0cf154094ad2a067793970eba"
               class="anim-state">
           <input type="radio" name="state" value="once" id="_anim_radio1_31395ce0cf154094ad2a067793970eba"
                  >
           <label for="_anim_radio1_31395ce0cf154094ad2a067793970eba">Once</label>
           <input type="radio" name="state" value="loop" id="_anim_radio2_31395ce0cf154094ad2a067793970eba"
                  checked>
           <label for="_anim_radio2_31395ce0cf154094ad2a067793970eba">Loop</label>
           <input type="radio" name="state" value="reflect" id="_anim_radio3_31395ce0cf154094ad2a067793970eba"
                  >
           <label for="_anim_radio3_31395ce0cf154094ad2a067793970eba">Reflect</label>
         </form>
       </div>
     </div>


     <script language="javascript">
       /* Instantiate the Animation class. */
       /* The IDs given should match those used in the template above. */
       (function() {
         var img_id = "_anim_img31395ce0cf154094ad2a067793970eba";
         var slider_id = "_anim_slider31395ce0cf154094ad2a067793970eba";
         var loop_select_id = "_anim_loop_select31395ce0cf154094ad2a067793970eba";
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     "


         /* set a timeout to make sure all the above elements are created before
            the object is initialized. */
         setTimeout(function() {
             anim31395ce0cf154094ad2a067793970eba = new Animation(frames, img_id, slider_id, 66.0,
                                      loop_select_id);
         }, 0);
       })()
     </script>






.. GENERATED FROM PYTHON SOURCE LINES 338-340

A frame from the animation.
sphinx_gallery_thumbnail_number = 4

.. GENERATED FROM PYTHON SOURCE LINES 340-344

.. code-block:: Python

    fig, ax, line1, line2, pfeil = init()
    update(4.15)

    plt.show()



.. image-sg:: /examples/beginner/images/sphx_glr_plot_light_diffraction_005.png
   :alt: running time 4.15 sec   Darker shade is slower maximum speed   The speed is the green arrow
   :srcset: /examples/beginner/images/sphx_glr_plot_light_diffraction_005.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 43.790 seconds)


.. _sphx_glr_download_examples_beginner_plot_light_diffraction.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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