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  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
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   "source": [
    "# Path following vs Trajectory Tracking MPC\n",
    "\n",
    "## Introduction\n",
    "\n",
    "In this example we will see how to define trajectory tracking and path following control problems using Neo.\n",
    "\n",
    "For this example we consider a point mass. The goal is to follow this shape by using a path-following approach.\n",
    "\n",
    "![Lissajous curve](../images/path_infty.png)\n",
    "\n",
    "the equation that describe it are $x(\\theta) = sin(\\theta)$ and $y(\\theta) = sin(2 \\theta)$.\n",
    "\n",
    "First import the necessary packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
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   "outputs": [],
   "source": [
    "# Add NMPC to path. NOT NECESSARY if it was installed via pip.\n",
    "import sys\n",
    "sys.path.append('../../../')\n",
    "\n",
    "# Import NMPC and Model classes\n",
    "from hilo_mpc import NMPC, Model\n",
    "import casadi as ca"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Build the model "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
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   "source": [
    "# Define the model\n",
    "model = Model(plot_backend='bokeh')\n",
    "\n",
    "# Constants\n",
    "M = 5.\n",
    "\n",
    "# States and algebraic variables\n",
    "xx = model.set_dynamical_states(['x', 'vx', 'y', 'vy'])\n",
    "model.set_measurements(['y_x', 'y_vx', 'y_y', 'y_vy'])\n",
    "model.set_measurement_equations([xx[0], xx[1], xx[2], xx[3]])\n",
    "x = xx[0]\n",
    "vx = xx[1]\n",
    "y = xx[2]\n",
    "vy = xx[3]\n",
    "\n",
    "# Inputs\n",
    "F = model.set_inputs(['Fx', 'Fy'])\n",
    "Fx = F[0]\n",
    "Fy = F[1]\n",
    "\n",
    "# ODEs\n",
    "dd1 = vx\n",
    "dd2 = Fx / M\n",
    "dd3 = vy\n",
    "dd4 = Fy / M\n",
    "\n",
    "model.set_dynamical_equations([dd1, dd2, dd3, dd4])\n",
    "\n",
    "# Initial conditions\n",
    "x0 = [0, 0, 0, 0]\n",
    "u0 = [0., 0.]\n",
    "\n",
    "# time interval\n",
    "dt = 0.1\n",
    "model.setup(dt=dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Path-following problem\n",
    "Once the model is setup, we can pass it to the path-following MPC. Defining a path-following problem can be done in  3 steps:\n",
    "\n",
    "1. Run `create_path_variable()` to generate the path variable\n",
    "2. Define the path function using the path variable\n",
    "3. Pass the path function function to the `ref` argument of stage and terminal cost and put `path_following=True`\n",
    "\n",
    "*NOTE*: you can create only as many path variables and you can have as many different path functions as you want."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "# Define path following MPC\n",
    "nmpc = NMPC(model)\n",
    "theta = nmpc.create_path_variable(u_pf_lb=1e-6)\n",
    "nmpc.quad_stage_cost.add_states(names=['x', 'y'], weights=[10, 10],\n",
    "                                ref=ca.vertcat(ca.sin(theta), ca.sin(2 * theta)), path_following=True)\n",
    "nmpc.quad_terminal_cost.add_states(names=['x', 'y'], weights=[10, 10],\n",
    "                                   ref=ca.vertcat(ca.sin(theta), ca.sin(2 * theta)), path_following=True)\n",
    "nmpc.horizon = 10\n",
    "nmpc.set_initial_guess(x_guess=x0, u_guess=u0)\n",
    "nmpc.setup(options={'print_level': 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "Then run the control loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "******************************************************************************\n",
      "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
      " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
      "         For more information visit http://projects.coin-or.org/Ipopt\n",
      "******************************************************************************\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Prepare and run closed loop\n",
    "n_steps = 200\n",
    "model.set_initial_conditions(x0=x0)\n",
    "sol = model.solution\n",
    "xt = x0.copy()\n",
    "for step in range(n_steps):\n",
    "    u = nmpc.optimize(xt)\n",
    "    model.simulate(u=u)\n",
    "    xt = sol['x:f']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "The results look like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
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       "     \"<div style='background-color: #fdd'>\\n\"+\n",
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       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
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       "     \"<ul>\\n\"+\n",
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Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"1002\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\": \"YF85VygJKMVnHE+lLv2AM93Vbstr0yo2TbIu5v8se5Rq3UQAUmcuh4aaJwNlpKwa\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\": \"KKuas3gevv3PvrlkyCMzffFeaMq5we/a2QsP5AUoS3mJ0jmaCL7jirFJN3GoE/lM\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\": \"MK/uFc3YT18pkvvXRl66tTHjP0/dxoSH2e/eiNMFIguKlun2+WVqaPTWmUy/zvh4\"};\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      if (url in hashes) {\n        element.crossOrigin = \"anonymous\";\n        element.integrity = \"sha384-\" + hashes[url];\n      }\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  \n  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\"];\n  var css_urls = [];\n  \n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    function(Bokeh) {\n    \n    \n    }\n  ];\n\n  function run_inline_js() {\n    \n    if (root.Bokeh !== undefined || force === true) {\n      \n    for (var i = 0; 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Application\",\"version\":\"2.3.1\"}};\n",
       "  var render_items = [{\"docid\":\"780cd0b6-b65f-4ba5-aa81-3571e437d9c6\",\"root_ids\":[\"1003\"],\"roots\":{\"1003\":\"e42613b7-6a19-409c-8a4d-940f61b3d525\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1003"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from bokeh.io import output_notebook, show\n",
    "from bokeh.plotting import figure, gridplot\n",
    "import numpy as np\n",
    "\n",
    "# define path function for plotting\n",
    "def path(theta):\n",
    "    return np.sin(theta), np.sin(2 * theta)\n",
    "x_path = []\n",
    "y_path = []\n",
    "for t in range(1000):\n",
    "    x_p, y_p = path(t / 100)\n",
    "    x_path.append(x_p)\n",
    "    y_path.append(y_p)\n",
    "\n",
    "# Plot using bokeh\n",
    "output_notebook()\n",
    "p_pf = figure(title='Path following problem')\n",
    "p_pf.line(x=np.array(model.solution['x']).squeeze(), y=np.array(model.solution['y']).squeeze())\n",
    "p_pf.line(x=x_path, y=y_path, line_color='red', line_dash='dashed')\n",
    "# p = format_figure(p)\n",
    "p_pf.yaxis.axis_label = \"y [m]\"\n",
    "p_pf.xaxis.axis_label = \"x [m]\"\n",
    "show(p_pf)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Trajectory tracking problem\n",
    "Trajectory tracking problems are formulated similarly to path following problem. For the case where the function of the\n",
    "trajectory is available, do as follows:\n",
    "\n",
    "1. Run `create_time_variable()` to generate the time variable\n",
    "2. Define the time-varying function using the time variable\n",
    "3. Pass the time function function to the `ref` argument of stage and terminal cost and put `trajectory_tracking=True`\n",
    "\n",
    "*NOTE*: if the function of the reference is not available, you can pass directly the data points when solving the MPC.\n",
    "see [the documentation](../modules/controllers.rst#trajectory-tracking-problem) for more info."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define trajectory tracking MPC\n",
    "nmpc = NMPC(model)\n",
    "time = nmpc.get_time_variable()\n",
    "nmpc.quad_stage_cost.add_states(names=['x', 'y'], weights=[10, 10],\n",
    "                                ref=ca.vertcat(ca.sin(time), ca.sin(2 * time)), trajectory_tracking=True)\n",
    "nmpc.quad_terminal_cost.add_states(names=['x', 'y'], weights=[10, 10],\n",
    "                                   ref=ca.vertcat(ca.sin(time), ca.sin(2 * time)), trajectory_tracking=True)\n",
    "nmpc.horizon = 10\n",
    "nmpc.set_initial_guess(x_guess=x0, u_guess=u0)\n",
    "nmpc.setup(options={'print_level': 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The rest is the same"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prepare and run closed loop\n",
    "n_steps = 100\n",
    "# the solution stored in the model must be reset (initial conditions are kept by default)\n",
    "model.reset_solution()\n",
    "sol = model.solution\n",
    "xt = x0.copy()\n",
    "for step in range(n_steps):\n",
    "    u = nmpc.optimize(xt)\n",
    "    model.simulate(u=u)\n",
    "    xt = sol['x:f']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    <div class=\"bk-root\">\n",
       "        <a href=\"https://bokeh.org\" target=\"_blank\" class=\"bk-logo bk-logo-small bk-logo-notebook\"></a>\n",
       "        <span id=\"1118\">Loading BokehJS ...</span>\n",
       "    </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "(function(root) {\n",
       "  function now() {\n",
       "    return new Date();\n",
       "  }\n",
       "\n",
       "  var force = true;\n",
       "\n",
       "  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n",
       "    root._bokeh_onload_callbacks = [];\n",
       "    root._bokeh_is_loading = undefined;\n",
       "  }\n",
       "\n",
       "  var JS_MIME_TYPE = 'application/javascript';\n",
       "  var HTML_MIME_TYPE = 'text/html';\n",
       "  var EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
       "  var CLASS_NAME = 'output_bokeh rendered_html';\n",
       "\n",
       "  /**\n",
       "   * Render data to the DOM node\n",
       "   */\n",
       "  function render(props, node) {\n",
       "    var script = document.createElement(\"script\");\n",
       "    node.appendChild(script);\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when an output is cleared or removed\n",
       "   */\n",
       "  function handleClearOutput(event, handle) {\n",
       "    var cell = handle.cell;\n",
       "\n",
       "    var id = cell.output_area._bokeh_element_id;\n",
       "    var server_id = cell.output_area._bokeh_server_id;\n",
       "    // Clean up Bokeh references\n",
       "    if (id != null && id in Bokeh.index) {\n",
       "      Bokeh.index[id].model.document.clear();\n",
       "      delete Bokeh.index[id];\n",
       "    }\n",
       "\n",
       "    if (server_id !== undefined) {\n",
       "      // Clean up Bokeh references\n",
       "      var cmd = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n",
       "      cell.notebook.kernel.execute(cmd, {\n",
       "        iopub: {\n",
       "          output: function(msg) {\n",
       "            var id = msg.content.text.trim();\n",
       "            if (id in Bokeh.index) {\n",
       "              Bokeh.index[id].model.document.clear();\n",
       "              delete Bokeh.index[id];\n",
       "            }\n",
       "          }\n",
       "        }\n",
       "      });\n",
       "      // Destroy server and session\n",
       "      var cmd = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n",
       "      cell.notebook.kernel.execute(cmd);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when a new output is added\n",
       "   */\n",
       "  function handleAddOutput(event, handle) {\n",
       "    var output_area = handle.output_area;\n",
       "    var output = handle.output;\n",
       "\n",
       "    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n",
       "    if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n",
       "      return\n",
       "    }\n",
       "\n",
       "    var toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
       "\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n",
       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
       "      // store reference to embed id on output_area\n",
       "      output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n",
       "    }\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
       "      var bk_div = document.createElement(\"div\");\n",
       "      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "      var script_attrs = bk_div.children[0].attributes;\n",
       "      for (var i = 0; i < script_attrs.length; i++) {\n",
       "        toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n",
       "        toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n",
       "      }\n",
       "      // store reference to server id on output_area\n",
       "      output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
       "    }\n",
       "  }\n",
       "\n",
       "  function register_renderer(events, OutputArea) {\n",
       "\n",
       "    function append_mime(data, metadata, element) {\n",
       "      // create a DOM node to render to\n",
       "      var toinsert = this.create_output_subarea(\n",
       "        metadata,\n",
       "        CLASS_NAME,\n",
       "        EXEC_MIME_TYPE\n",
       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
       "      // Render to node\n",
       "      var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
       "      return toinsert\n",
       "    }\n",
       "\n",
       "    /* Handle when an output is cleared or removed */\n",
       "    events.on('clear_output.CodeCell', handleClearOutput);\n",
       "    events.on('delete.Cell', handleClearOutput);\n",
       "\n",
       "    /* Handle when a new output is added */\n",
       "    events.on('output_added.OutputArea', handleAddOutput);\n",
       "\n",
       "    /**\n",
       "     * Register the mime type and append_mime function with output_area\n",
       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "      /* Is output safe? */\n",
       "      safe: true,\n",
       "      /* Index of renderer in `output_area.display_order` */\n",
       "      index: 0\n",
       "    });\n",
       "  }\n",
       "\n",
       "  // register the mime type if in Jupyter Notebook environment and previously unregistered\n",
       "  if (root.Jupyter !== undefined) {\n",
       "    var events = require('base/js/events');\n",
       "    var OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  \n",
       "  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n",
       "    root._bokeh_timeout = Date.now() + 5000;\n",
       "    root._bokeh_failed_load = false;\n",
       "  }\n",
       "\n",
       "  var NB_LOAD_WARNING = {'data': {'text/html':\n",
       "     \"<div style='background-color: #fdd'>\\n\"+\n",
       "     \"<p>\\n\"+\n",
       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
       "     \"</p>\\n\"+\n",
       "     \"<ul>\\n\"+\n",
       "     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
       "     \"</ul>\\n\"+\n",
       "     \"<code>\\n\"+\n",
       "     \"from bokeh.resources import INLINE\\n\"+\n",
       "     \"output_notebook(resources=INLINE)\\n\"+\n",
       "     \"</code>\\n\"+\n",
       "     \"</div>\"}};\n",
       "\n",
       "  function display_loaded() {\n",
       "    var el = document.getElementById(\"1118\");\n",
       "    if (el != null) {\n",
       "      el.textContent = \"BokehJS is loading...\";\n",
       "    }\n",
       "    if (root.Bokeh !== undefined) {\n",
       "      if (el != null) {\n",
       "        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n",
       "      }\n",
       "    } else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(display_loaded, 100)\n",
       "    }\n",
       "  }\n",
       "\n",
       "\n",
       "  function run_callbacks() {\n",
       "    try {\n",
       "      root._bokeh_onload_callbacks.forEach(function(callback) {\n",
       "        if (callback != null)\n",
       "          callback();\n",
       "      });\n",
       "    } finally {\n",
       "      delete root._bokeh_onload_callbacks\n",
       "    }\n",
       "    console.debug(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(css_urls, js_urls, callback) {\n",
       "    if (css_urls == null) css_urls = [];\n",
       "    if (js_urls == null) js_urls = [];\n",
       "\n",
       "    root._bokeh_onload_callbacks.push(callback);\n",
       "    if (root._bokeh_is_loading > 0) {\n",
       "      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
       "      return null;\n",
       "    }\n",
       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "    root._bokeh_is_loading = css_urls.length + js_urls.length;\n",
       "\n",
       "    function on_load() {\n",
       "      root._bokeh_is_loading--;\n",
       "      if (root._bokeh_is_loading === 0) {\n",
       "        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n",
       "        run_callbacks()\n",
       "      }\n",
       "    }\n",
       "\n",
       "    function on_error(url) {\n",
       "      console.error(\"failed to load \" + url);\n",
       "    }\n",
       "\n",
       "    for (let i = 0; i < css_urls.length; i++) {\n",
       "      const url = css_urls[i];\n",
       "      const element = document.createElement(\"link\");\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error.bind(null, url);\n",
       "      element.rel = \"stylesheet\";\n",
       "      element.type = \"text/css\";\n",
       "      element.href = url;\n",
       "      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n",
       "      document.body.appendChild(element);\n",
       "    }\n",
       "\n",
       "    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\": \"YF85VygJKMVnHE+lLv2AM93Vbstr0yo2TbIu5v8se5Rq3UQAUmcuh4aaJwNlpKwa\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\": \"KKuas3gevv3PvrlkyCMzffFeaMq5we/a2QsP5AUoS3mJ0jmaCL7jirFJN3GoE/lM\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\": \"MK/uFc3YT18pkvvXRl66tTHjP0/dxoSH2e/eiNMFIguKlun2+WVqaPTWmUy/zvh4\"};\n",
       "\n",
       "    for (let i = 0; i < js_urls.length; i++) {\n",
       "      const url = js_urls[i];\n",
       "      const element = document.createElement('script');\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error.bind(null, url);\n",
       "      element.async = false;\n",
       "      element.src = url;\n",
       "      if (url in hashes) {\n",
       "        element.crossOrigin = \"anonymous\";\n",
       "        element.integrity = \"sha384-\" + hashes[url];\n",
       "      }\n",
       "      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n",
       "      document.head.appendChild(element);\n",
       "    }\n",
       "  };\n",
       "\n",
       "  function inject_raw_css(css) {\n",
       "    const element = document.createElement(\"style\");\n",
       "    element.appendChild(document.createTextNode(css));\n",
       "    document.body.appendChild(element);\n",
       "  }\n",
       "\n",
       "  \n",
       "  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\"];\n",
       "  var css_urls = [];\n",
       "  \n",
       "\n",
       "  var inline_js = [\n",
       "    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "    function(Bokeh) {\n",
       "    \n",
       "    \n",
       "    }\n",
       "  ];\n",
       "\n",
       "  function run_inline_js() {\n",
       "    \n",
       "    if (root.Bokeh !== undefined || force === true) {\n",
       "      \n",
       "    for (var i = 0; i < inline_js.length; i++) {\n",
       "      inline_js[i].call(root, root.Bokeh);\n",
       "    }\n",
       "    if (force === true) {\n",
       "        display_loaded();\n",
       "      }} else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(run_inline_js, 100);\n",
       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "      root._bokeh_failed_load = true;\n",
       "    } else if (force !== true) {\n",
       "      var cell = $(document.getElementById(\"1118\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\n",
       "\n",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.bokehjs_load.v0+json": "\n(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  var force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n  \n\n  \n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  var NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"1118\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\": \"YF85VygJKMVnHE+lLv2AM93Vbstr0yo2TbIu5v8se5Rq3UQAUmcuh4aaJwNlpKwa\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\": \"KKuas3gevv3PvrlkyCMzffFeaMq5we/a2QsP5AUoS3mJ0jmaCL7jirFJN3GoE/lM\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\": \"MK/uFc3YT18pkvvXRl66tTHjP0/dxoSH2e/eiNMFIguKlun2+WVqaPTWmUy/zvh4\"};\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      if (url in hashes) {\n        element.crossOrigin = \"anonymous\";\n        element.integrity = \"sha384-\" + hashes[url];\n      }\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  \n  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.3.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.3.1.min.js\"];\n  var css_urls = [];\n  \n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    function(Bokeh) {\n    \n    \n    }\n  ];\n\n  function run_inline_js() {\n    \n    if (root.Bokeh !== undefined || force === true) {\n      \n    for (var i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\n    if (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(\"1118\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"37a6fdd6-16b8-4d46-baf6-2429e4c5da35\" data-root-id=\"1119\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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problem\"},\"id\":\"1120\",\"type\":\"Title\"},{\"attributes\":{\"data_source\":{\"id\":\"1152\"},\"glyph\":{\"id\":\"1153\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"1154\"},\"view\":{\"id\":\"1156\"}},\"id\":\"1155\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"data\":{\"x\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[101]},\"y\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"order\":\"little\",\"shape\":[101]}},\"selected\":{\"id\":\"1182\"},\"selection_policy\":{\"id\":\"1181\"}},\"id\":\"1152\",\"type\":\"ColumnDataSource\"},{\"attributes\":{},\"id\":\"1175\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1131\",\"type\":\"BasicTicker\"},{\"attributes\":{},\"id\":\"1177\",\"type\":\"AllLabels\"},{\"attributes\":{\"data_source\":{\"id\":\"1157\"},\"glyph\":{\"id\":\"1158\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"1159\"},\"view\":{\"id\":\"1161\"}},\"id\":\"1160\",\"type\":\"GlyphRenderer\"},{\"attributes\":{},\"id\":\"1183\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1184\",\"type\":\"Selection\"},{\"attributes\":{\"overlay\":{\"id\":\"1144\"}},\"id\":\"1140\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1153\",\"type\":\"Line\"},{\"attributes\":{\"source\":{\"id\":\"1157\"}},\"id\":\"1161\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"1182\",\"type\":\"Selection\"},{\"attributes\":{\"active_multi\":null,\"tools\":[{\"id\":\"1138\"},{\"id\":\"1139\"},{\"id\":\"1140\"},{\"id\":\"1141\"},{\"id\":\"1142\"},{\"id\":\"1143\"}]},\"id\":\"1145\",\"type\":\"Toolbar\"},{\"attributes\":{\"line_color\":\"red\",\"line_dash\":[6],\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1158\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1124\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"1178\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1122\",\"type\":\"DataRange1d\"},{\"attributes\":{\"source\":{\"id\":\"1152\"}},\"id\":\"1156\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"1128\",\"type\":\"LinearScale\"},{\"attributes\":{\"axis_label\":\"y [m]\",\"formatter\":{\"id\":\"1178\"},\"major_label_policy\":{\"id\":\"1180\"},\"ticker\":{\"id\":\"1135\"}},\"id\":\"1134\",\"type\":\"LinearAxis\"},{\"attributes\":{\"axis\":{\"id\":\"1134\"},\"dimension\":1,\"ticker\":null},\"id\":\"1137\",\"type\":\"Grid\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1154\",\"type\":\"Line\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"red\",\"line_dash\":[6],\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"1159\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"1181\",\"type\":\"UnionRenderers\"}],\"root_ids\":[\"1119\"]},\"title\":\"Bokeh Application\",\"version\":\"2.3.1\"}};\n",
       "  var render_items = [{\"docid\":\"6272467d-507c-487a-a918-afe4b783d7c7\",\"root_ids\":[\"1119\"],\"roots\":{\"1119\":\"37a6fdd6-16b8-4d46-baf6-2429e4c5da35\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1119"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_path = []\n",
    "y_path = []\n",
    "for t in range(1000):\n",
    "    x_p, y_p = path(t / 100)\n",
    "    x_path.append(x_p)\n",
    "    y_path.append(y_p)\n",
    "\n",
    "# plot using bokeh\n",
    "output_notebook()\n",
    "p_tt = figure(title='Trajectory tracking problem')\n",
    "p_tt.line(x=np.array(model.solution['x']).squeeze(), y=np.array(model.solution['y']).squeeze())\n",
    "p_tt.line(x=x_path, y=y_path, line_color='red', line_dash='dashed')\n",
    "# p = format_figure(p)\n",
    "p_tt.yaxis.axis_label = \"y [m]\"\n",
    "p_tt.xaxis.axis_label = \"x [m]\"\n",
    "show(p_tt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"3d24eb5e-43b0-4bc3-86ac-191d15bbed1d\" data-root-id=\"1375\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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following problem\"},\"id\":\"1004\",\"type\":\"Title\"},{\"attributes\":{},\"id\":\"1047\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{},\"id\":\"1053\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1054\",\"type\":\"Selection\"},{\"attributes\":{\"source\":{\"id\":\"1152\"}},\"id\":\"1156\",\"type\":\"CDSView\"},{\"attributes\":{\"axis\":{\"id\":\"1134\"},\"dimension\":1,\"ticker\":null},\"id\":\"1137\",\"type\":\"Grid\"},{\"attributes\":{\"active_multi\":null,\"tools\":[{\"id\":\"1022\"},{\"id\":\"1023\"},{\"id\":\"1024\"},{\"id\":\"1025\"},{\"id\":\"1026\"},{\"id\":\"1027\"}]},\"id\":\"1029\",\"type\":\"Toolbar\"},{\"attributes\":{},\"id\":\"1055\",\"type\":\"UnionRenderers\"}],\"root_ids\":[\"1375\"]},\"title\":\"Bokeh Application\",\"version\":\"2.3.1\"}};\n",
       "  var render_items = [{\"docid\":\"d52a5a2f-3f37-4c51-bba1-e7f9fc80aede\",\"root_ids\":[\"1375\"],\"roots\":{\"1375\":\"3d24eb5e-43b0-4bc3-86ac-191d15bbed1d\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1375"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from bokeh.models import Range1d\n",
    "\n",
    "p_pf.x_range = Range1d(-1,-0.5)\n",
    "p_pf.y_range = Range1d(0.5, 1)\n",
    "\n",
    "p_tt.x_range = Range1d(-1,-0.5)\n",
    "p_tt.y_range = Range1d(0.5, 1)\n",
    "\n",
    "show(gridplot([p_pf,p_tt], ncols=2))"
   ]
  }
 ],
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