Motion_Models.ipynb

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      "name": "python",
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      "pygments_lexer": "ipython3",
      "version": "3.5.2"
    },
    "colab": {
      "name": "Motion Models.ipynb",
      "provenance": [],
      "collapsed_sections": []
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BnbmGWo_JbYd"
      },
      "source": [
        "# Exercise 4\n",
        "\n",
        "**_DISCLAIMER:_** The notation used in this exercise follows the one of the Probabilistic Robotics book (refer to Chapter 5.4 in case you have doubts).\n",
        "\n",
        "## 4.1 Inverse motion model\n",
        "\n",
        "The odometry model uses the _relative motion information_. The odometry readings are $u_t = [{\\overline{x}}_{t-1} , {\\overline{x}}_{t}]$, where $\\overline{x}_{t-1}$ and  $\\overline{x}_t$ are poses in a robot-internal coordinate frame (different from the map).\n",
        "\n",
        "The function `inverse_motion_model` takes as input an odometry reading $u_t$ that consist in:\n",
        "\n",
        "- the initial pose of the robot in the odometry coordinate frame $\\overline{x}_{t-1} = [\\overline{x},\\overline{y},\\overline{\\theta}]$\n",
        "- the estimated pose of the robot in the odometry coordinate frame $\\overline{x}_t = [\\overline{x}',\\overline{y}',\\overline{\\theta}']$\n",
        "\n",
        "The output is the relative motion $\\delta_{rot1}, \\delta_{trans}, \\delta_{rot2}$.\n",
        "\n",
        "Implement the function `inverse_motion_model` and verify that it is correct for some test input. **[2.0]**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JJRiYnG6JbYq"
      },
      "source": [
        "from ex7 import *\n",
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "aPSLpOP2yr0B",
        "outputId": "787b751c-9e44-4ecb-e796-b43fa4447e37"
      },
      "source": [
        "inv_motion_model([[0,0,0],[1,0,0]])"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(0.0, 1.0, 0.0)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 2
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "46H3LomFJbYr"
      },
      "source": [
        "## 4.2 Odometry-based motion model\n",
        "\n",
        "The function `motion_model_odometry` computes the posterior $p(x_t | u_t, x_{t-1})$ from odometry readings.\n",
        "\n",
        "This function takes as input:\n",
        "\n",
        "- the initial pose of the robot $x_{t-1} = [x,y,\\theta]$ _(**map** coordinate frame)_\n",
        "- the hypothesized (or query) final pose $x_{t} = [x', y', \\theta']$ _(**map** coordinate frame)_\n",
        "- the odometry readings $u_t = [\\overline{x}_{t-1} \\overline{x}_t]$ _(**odometry** coordinate frame)_\n",
        "- the noise parameters $\\mathbf{\\alpha} = [\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4]$\n",
        "\n",
        "The output is the probability $p(x_t | u_t, x_{t-1})$\n",
        "\n",
        "Assume that a robot starts at pose $x_0 = [2.0, 3.0, 0.0]$ in the map frame and moves according to a motion model with $\\mathbf{\\alpha} = [1.0, 1.0, 0.01, 0.01]$.\n",
        "\n",
        "The robot excecutes one motion command and the odometry readings are:\n",
        "\n",
        "1. $\\overline{x}_0 = [0.0 , 0.0 , 0.0   ]$\n",
        "2. $\\overline{x}_1 = [0.5 , 0.0 , \\pi/2 ]$\n",
        "\n",
        "Implement the `motion_model_odometry` function and verify that it is correct for some test input. **[1.0]**\n",
        "\n",
        "---\n",
        "\n",
        "Consider a 150x150 grid map the world with a resolution of 0.01, centered in the original position of the robot.\n",
        "\n",
        "Plot the posterior $p(x_t | u_t, x_{t-1})$ for all possible $[x, y]$ values from the grid. **[2.0]**\n",
        "\n",
        "**Note that** the query input is a position, not a pose. Therefore, to plot the posterior belief over the gridmap, you can assume the term $\\hat{\\delta}_\\mathrm{rot2}$ to be zero and, for each position, integrate over all possible orientations. This can be implemented by considering $p_3 = 1.0$ in the equations. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "u-Ii-YeaJbYs"
      },
      "source": [
        "# Since we are using triangular normal distribution we get the following plot as the result \n",
        "# and at 1 it is at peak and hence we can verify it that motion model is correct\n",
        "\n",
        "u = [[0.0,0.0,0.0],[1.0,0.0,0.0]]\n",
        "x_0 = [0.0,0.0,0.0]\n",
        "alpha = [1.0,1.0,0.01,0.01]\n",
        "p = []\n",
        "x = []\n",
        "for i in range(50):\n",
        "    if i<=25:\n",
        "\n",
        "        query = [1-(0.01)*i,0.0,0.0]\n",
        "        prob = motion_model_odometry(x_0, query, u, alpha, gridmap = True)\n",
        "        p.append(prob)\n",
        "        x.append(query[0])\n",
        "    else:\n",
        "        query = [1+(0.01)*(i-25),0.0,0.0]\n",
        "        prob = motion_model_odometry(x_0, query, u, alpha, gridmap = True)\n",
        "        p.append(prob)\n",
        "        x.append(query[0])"
      ],
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "5CiEMtDtzh28",
        "outputId": "f47acd6f-73ed-4a89-8e91-56dc13dbbfa9"
      },
      "source": [
        "plt.scatter(x, p)\n",
        "plt.show()"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OARmpmDJJbYt"
      },
      "source": [
        "grid_map = np.zeros((150,150))\n",
        "u_t      = [[0.0,0.0,0.0],[0.5,0.0,math.pi/2]]\n",
        "x_init   = [2.0,3.0,0.0]\n",
        "alpha    = [1.0,1.0,0.01,0.01]\n",
        "\n",
        "for i in range(grid_map.shape[0]):\n",
        "    for j in range(grid_map.shape[1]):\n",
        "        grid_map[i][j] = motion_model_odometry(x_init, map2world(i, j, x_init,grid_map), u_t, alpha,gridmap = True)"
      ],
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "IPLHQoeqz3U7",
        "outputId": "a3d434e1-b581-4f11-c811-bfb89c147af4"
      },
      "source": [
        "plot_gridmap(grid_map)"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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s0G63y1hBtHQzMzOlSxGfTVusg+wmpGZ/Ouw3fP8QgYgBxP7y8jLr6+sANBoN5ubmSlGNGEBYBum1zMEFSJEIDJi0rz9dYyCsAzMrXQMz4+zZs6ytrVGr1VhfXy8j12HyhhUQ29WH+KCNLUjnZKQz/dJI/+rqahljWVlZKSv/6uoqALOzsxw+fJjZ2dnylQYCq12CuSERGAJpK51W1AhGpf3UU1NTXLhwgZWVFU6fPs3c3BzNZpO5ublSEJrNZmnKTk9Pl61bvV7f5iZMctygOtgnfPzo7ltZWSlFYGVlpextWV5eZmVlBTNjfn6eo0ePMj8/X16/uFZzc3NlDCCuWa6jNPey2vD9wDuBM+7+48Wx3wR+GmgB3wHe7+7ninP3AncCm8AvufufDKjsE8fU1FRZ6SMmkC47FvGBOLe1tXVRaxcPcAQPwypotVrU6/VSJGJdg/iNcR9rkKb6Sgf7hK+fjvJLRWB1dbWs+NHbEhV7YWGBQ4cOMTMzw9zcXCmc3SyAg2Q9XSp7sQQ+BfwX4DPJsYeAe929bWb3AfcCHzazm4A7gL8L/BDwp2b2Y+6+2d9iTybpfPR0jYF04kp0E9brdba2tjh79uy2h77ZbJbmbvi2cTysgvX19a4BxG6+7qge/vT/T7tI0wU+IriXCmFsLy0tlUHA5eXlcgJQrVZjYWGB+fl5jhw5UsYBwgUIEUgHA+VqAQR7WXL8a2Z2rHLsK8nuI8C7i+0TwOfcfR34KzM7BdwC/HlfSnsASFeqScelp5NVGo3GtlZreXmZCxcucPbsWaAjJvPz86VbMD8/X0a4Z2dny8ofVkG4GmEhpMHEYfvBaSsfgb5o6dPxE9XBPlHZI/AX29H6Hzp0iPn5eRYXF1lcXGRubq50AaJnZW5urmz9m81meT9ypx8xgV8APl9sX0NHFIKJTkg6SNKVhWI/XIWpqamym8rMylYeKCvJuXPnSsGI1q7RaJT+byoC1UEx8bmIQaS/220l3bSM6d8gzbeQ5l+s5mVMl2SPQGc10h9R/na7XfaWRJAvtsNCCHE7fPhwafovLi6Wvv/s7Gx5LeL6yPy/mJ5EwMw+ArSBB/bx2buAuwCuv/76XooxsaRz09Mly6pdi/Hgbm5usry8XI4rSFvLqOzLy8ul2ZtWgrAEwgxOexRCGEIQopJUux136jpLK36ajamazy9a+2qkP13ZJ+3nj4oflX51dXXbyMoQtyNHjpRjABYWFkpRTC2j1NLK3fyvsm8RMLOfpxMwvM1/4OBdUkJS4BMAx48fzzpjZ7Uyhm8frff6+nr5UIdrMDs7y/LyMsvLy7z22muleMT76vV6uR3fl37n7Ozstt9N+8nTmXPhKuw2BqHa4lcX80xFIF3Np+rrV0f8xVDfjY2N8v9Lo/7z8/NcdtllpXsUg4Ci9Q8XKKwecTH7EgEzux34FeAfu/tKcupB4PfM7KN0AoM3Av+v51JmQLS8sb59WinTYFbEACJWsLS0RKPRKAfIrK2tcf78eYBtw2FDCCJynsYK0uGy6Rz6N1pZN11BOfZTNyAd2ZcG/Kqr+VS3qzGBKPPc3Fzp6szPz3P55ZeXrX4cD+snhC/E7aAOouoX+01Iei/QBB4qLu4j7v6v3f1pM/sC8AwdN+Fu9QzsnXSVorTypQtahDjAD2bBTU1NldHyGEYbre/6+vo2gQkRSCfMpGMU0gk0qSWQikB1WbTq8eqkqHSGX2oJpEt8x7lqWvBarUaz2WRhYYHFxcXS5D969Ghp3UTrHyKZzgKU/787lnbVjIrjx4/7yZMnR12MsSSi39V+8qWlpbL1f/3111laWirnHywtLZUj5y5cuFBOnonx8Wnlr8YEqq7AThU9XV69aiFU4wE7pflOxaHVamFmHDp0qAzsLSwscPjw4TLKHyIQ58K9iRhANeAptmNmj7n78epxjRgcc6otdjrLMEzm2dnZbYIQIw6XlpZKC2FlZaX0r1utFufPny8DcmkcICrQXip4dY3FbkLRLTDo7mVlnp6eLrvxms1mKQKxnYpA+P3R3VddBCQNaIq9IxEYc1J/PO3Sm5qaotVqlQ9/LGgaJvHq6mo5Si5EIKyH1dXVi3ogNjY2cHdWV1e3RferI/iiErdarW1jDtJEKmllTAcppWVPJ/McOnSo9Omjq292dra0BKLiRywkApvpcmC5zP0fBBKBCSFEIBYmiYofS5NFqx6BslarxeLiIhcuXCiH1l64cKEMHsYU25hzH9sxCCe65NLx+ulCqe12u6zkEVyMMqVmelT22I4WP53Ku7CwUE7sSd2BtMWvzvyLl4J+vSMRmEBi/ULotOIhBhEI7LayTkytjYqdxhTSvvgQg+oS3XE+ndATIpAuyBE9EhGATNdAiNY+BjWlIhCVPZ3hl34+7edXq99fJAITSnWkYUw4isoXvQQzMzPbYgfpLLyYlBQz8FIhSCfvVAN54SakJn7axZi6KOmIxRCBnfrz0zX+0sFNEaNIRzaK/iEROACk4/9jJN3m5mY5wzAdkltdgz8qflUE0v77NKKfBvpSvz/tYgxfPR2uGyIQ57ut7psm/agm/BCDQyJwwEjXKYhJMmlQr5qQM+2zr07mifNpX39qCaQBwOqIw9iOwU+pOKQWQ3VBj3RdP5n7w0EicEBJK1A6CCgi/81mc1vUP63kMcMv7Q1Ix+yng4XSyH86KzHdTlv1biv57tQlKYaDRCAD0sqW0m0hj3hV06ynx2H7sOGqGKQjHKuJPKtzEsTokQhkTJo6rRvdRpPuNMK0WwuuVn0ykAiIHdmpYleFQJV9spEIiEtGlf5gob4XITJHIiBE5kgEhMgciYAQmSMRECJzJAJCZI5EQIjMkQgIkTm7ioCZ3W9mZ8zsqS7nftnM3MyuKPbNzH7bzE6Z2bfM7OZBFFoI0T/2Ygl8Cri9etDMrgN+CvhecvjtdHIN3Egnu9DHey+iEGKQ7CoC7v414LUupz5GJwFJOpD8BPAZ7/AIsGhmV/elpEKIgbCvmICZnQBecvdvVk5dA3w/2VdCUiHGnEueQGRmc8Cv0nEF9o0SkgoxHuzHEvgR4Abgm2b2XTpJRx83s7/NJSYkdffj7n78yiuv3EcxhBD94JJFwN2fdPe/5e7H3P0YHZP/Znd/mU5C0p8regluBV5399P9LbIQop/spYvws8CfA282sxfN7M43ePuXgeeBU8B/A/5NX0ophBgYu8YE3P29u5w/lmw7cHfvxRJCDAuNGBQicyQCQmSORECIzJEICJE5EgEhMkciIETmSASEyByJgBCZIxEQInNspwSTQy2E2avAMvDXoy5LwhWoPLsxbmVSed6YH3b3i2brjYUIAJjZSXc/PupyBCrP7oxbmVSe/SF3QIjMkQgIkTnjJAKfGHUBKqg8uzNuZVJ59sHYxASEEKNhnCwBIcQIGLkImNntZvaXRcKSe0ZUhuvM7Ktm9oyZPW1mHyiO/7qZvWRmTxSvdwyxTN81syeL3z1ZHLvMzB4ys+eKv0eHVJY3J9fgCTM7b2YfHPb16ZYIZ6drMoxEODuU5zfN7C+K3/wDM1ssjh8zs9XkWv1Ov8uzb9x9ZC+gBnwHeBPQAL4J3DSCclxNZ51EgEPAt4GbgF8H/v2Irs13gSsqx/4DcE+xfQ9w34ju2cvADw/7+gBvA24GntrtmgDvAP4YMOBW4NEhleengHqxfV9SnmPp+8bpNWpL4BbglLs/7+4t4HN0EpgMFXc/7e6PF9sXgGcZz3wJJ4BPF9ufBn5mBGW4DfiOu78w7B/27olwdromA0+E06087v4Vd28Xu4/QWXF7rBm1CIxdshIzOwa8BXi0OPSLhWl3/7DM7wIHvmJmjxU5GgCu8h+s3vwycNUQyxPcAXw22R/V9Ql2uibj8Gz9Ah1rJLjBzL5hZv/HzP7RkMuyI6MWgbHCzBaA3wc+6O7n6eRS/BHg7wOngf80xOL8hLvfTCe/491m9rb0pHdszKF27ZhZA3gX8D+LQ6O8PhcximuyE2b2EaANPFAcOg1c7+5vAT4E/J6ZHR5V+VJGLQJ7TlYyaMxsmo4APODuXwJw91fcfdPdt+gsoX7LsMrj7i8Vf88Af1D89ith0hZ/zwyrPAVvBx5391eKso3s+iTsdE1G9myZ2c8D7wR+thAm3H3d3f+m2H6MTizsx4ZRnt0YtQh8HbjRzG4oWpk76CQwGSpmZsAngWfd/aPJ8dSH/BfARenZB1SeeTM7FNt0gk1P0bk27yve9j7gD4dRnoT3krgCo7o+FXa6JiNJhGNmt9NJ1Psud19Jjl9pZrVi+010Mnc/P+jy7IlRRybpRHG/TUcZPzKiMvwEHTPyW8ATxesdwP8AniyOPwhcPaTyvIlOT8k3gafjugCXAw8DzwF/Clw2xGs0D/wNcCQ5NtTrQ0eATgMbdHz8O3e6JnR6Bf5r8Vw9CRwfUnlO0YlFxHP0O8V7/2VxL58AHgd+ehTPereXRgwKkTmjdgeEECNGIiBE5kgEhMgciYAQmSMRECJzJAJCZI5EQIjMkQgIkTn/H2MqrJzOfS9KAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Gqi2344RJbYt"
      },
      "source": [
        "## 4.3 Sample odometry motion model\n",
        "\n",
        "The `motion_model_odometry` requires high computation complexity and does not scale well to large real-world environments. \n",
        "\n",
        "One effective approach to approximate $p(x_t | u_t, x_{t-1})$ is to use **sampling**.\n",
        "\n",
        "The `sample_motion_model_odometry` function defines the sampling-based odometry motion model. \n",
        "\n",
        "This function takes as input:\n",
        "\n",
        "- the initial pose of the robot $x_{t-1} = [x,y,\\theta]$ _(**map** coordinate frame)_\n",
        "- the odometry readings $u_t = [\\overline{x}_{t-1} \\overline{x}_t]$ _(**odometry** coordinate frame)_\n",
        "- the noise parameters $\\mathbf{\\alpha} = [\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4]$\n",
        "\n",
        "The output is a new (sampled) pose predicted by the motion model.\n",
        "\n",
        "Implement the `sample_motion_model_odometry` function and verify that it is correct for some test input. **[2.0]**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "thv1p6ky0ZCD"
      },
      "source": [
        "# Here when we sample 1000 times we get the value of x as the approximation of the distribution \n",
        "# we are using hence we can verify that sample motion_model_odometry is correct\n",
        "\n",
        "x = np.array([0.0,0.0,0.0])\n",
        "alpha = np.array([0.1,0.1,0.01,0.01])\n",
        "u = np.array([[0.0,0.0,0.0],[1.0,0.0,0.0]])\n",
        "x_n= []\n",
        "for i in range(1000):\n",
        "    x_new,y_new,z_new = sample_motion_model(x, u, alpha)\n",
        "    x_n.append(x_new)\n",
        "x_n = np.array(x_n)"
      ],
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "0P19i8ia0thJ",
        "outputId": "5b7fed58-54ad-4c22-a5c6-eb260f2ca065"
      },
      "source": [
        "plt.hist(x_n)\n",
        "plt.show()"
      ],
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IFWyG1PYJbYu"
      },
      "source": [
        "## 4.4 Evaluate sample odometry motion model\n",
        "\n",
        "Assume that a robot starts at pose $x_0 = [2.0, 3.0, 0.0]$ in the map frame and moves according to a motion model with $\\mathbf{\\alpha} = [0.1, 0.1, 0.01, 0.01]$.\n",
        "\n",
        "The robot obtains the following odometry readings:\n",
        "\n",
        "1. $\\overline{x}_0 = [0.0 , 0.0 , 0.0   ]$\n",
        "2. $\\overline{x}_1 = [0.5 , 0.0 , \\pi/2 ]$\n",
        "3. $\\overline{x}_2 = [0.5 , 0.5 , 0.0   ]$\n",
        "4. $\\overline{x}_3 = [1.0 , 0.5 , 0.0   ]$\n",
        "5. $\\overline{x}_4 = [1.0 , 1.5 , \\pi/2 ]$\n",
        "6. $\\overline{x}_5 = [1.0 , 2.5 , \\pi/2 ]$\n",
        "\n",
        "Evaluate the `sample_motion_model_odometry` by considering 1000 samples and plot the resulting positions for each sample in one unique plot. **[3.0]**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HrcDosljJbYu"
      },
      "source": [
        "commands = np.array([[0.0,0.0,0.0],[0.5,0.0,math.pi/2],[0.5,0.5,0.0],[1.0,0.5,0.0],[1.0,1.5,math.pi/2],[1.0,2.5,math.pi/2]])\n",
        "alpha = np.array([0.1,0.1,0.01,0.01])\n",
        "x_initials = np.array([0.0,0.0,0.0])\n",
        "n_samples = 1000\n",
        "\n",
        "samples = np.array([2.0,3.0,0.0]*1000).reshape((1000,3))\n",
        "\n",
        "for i in range(1,len(commands)):\n",
        "        \n",
        "        x_init = np.array([2.0,3.0,0.0]) + np.array(commands[i-1])\n",
        "        \n",
        "        x_initials = np.vstack((x_initials,x_init))\n",
        "        \n",
        "        count = (i-1) * 1000\n",
        "        \n",
        "        u_t = np.array([commands[i-1], commands[i]])\n",
        "      \n",
        "        for j in range(1000):\n",
        "                 \n",
        "                 x_new,y_new,theta_new = sample_motion_model(samples[count + j],u_t,alpha)\n",
        "                 pose = np.array([x_new,y_new,theta_new])\n",
        "                 samples = np.vstack((samples,pose.T))\n",
        "              \n",
        "\n",
        "x_initials = np.vstack((x_initials,[3.0,5.5,0.0]))"
      ],
      "execution_count": 13,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "mFfOJHhl04c9",
        "outputId": "df592745-fc86-413c-fe05-8fa15abbcb18"
      },
      "source": [
        "# Green line showing the true path the robot has taken if there is no noise\n",
        "\n",
        "plt.plot(x_initials[1:,0], x_initials[1:,1], \"--g\")\n",
        "plt.scatter(samples[:,0], samples[0:,1])\n",
        "plt.show()"
      ],
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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pO2sowiXEoLvA0GjWdky78kI0u1hnp5S6C4tmYY7KMdXzjM2ms/uPjlctqFV+vvvaK2SBKgCsHoxWHrwdCWoYGXnrNPp3HzGcgUqjDHPEoDeJ5uXaQX8hblq9yDDGTFTfQ6rnodjxjK2ms2afbwWP2w/cVG7U8+CNdO5hV72YhS/TKQX9a22otFoUMehN4qTKoVHpE/0tVYA9nfb5i9PleiWNItPZYPBCuWH1AAZqjb1ZYhUArPjlK3DgjdMNjcNrZs1oE2NugRj0JnESozw/la+6cbc+fQR5nVIlX2BLJYxGZYf1Ri9wv7TYQjV+KzeMjL1ZKKZDSeCnJ8Lb8FoWQ62xlSlKRMeJ6DARjRFRTZshKvJfiejnRPQaEf2a+0MNJ0692Uo5mZknXmDUZPkZ5W02W/PCrXolrc6Wm7Zgy01bbO8fBuWGUVYyodj8I+gFUStk9miNEw99JTOfMtn26wB+pfTvRgD/o/R/rBkazTbU1NnOjTtTSWBGW6LcqstsYatZIyAx8ea55eO3ONo/DKEufd0er1sOusX5i9PYMnQY+4+Oy8K8AW7VcvkNAH/LRYYBpIlonkvHDiVaHLSRps7ajWuVMXlmUsXF6QIeXt9ZLq1rdSwhOMbeG8PYe+YxaT12a/Z4jVbi16hsc1iZyKnYOXyiqtxu7+AYtgzZEybEHbsGnQE8T0QHiWijwfYMgJMVr98uvVcFEW0kohEiGhkft6fbDiuNtnyrvHH71y6xbBJRGVIJixEQaul9rhe9z/Xa3j9soS6/Qj1aUp0X7Bw+IUYd9kMun2HmLBF9FMA/ENFRZn7J6Zcx8/cAfA8Auru7o+IUGNLoTVB549opV6u972btESF4whTqarQ2kBNmtReLwt3/2KEaIYBb7Bo+UdUkvBWx5aEzc7b0//sAngJwg26XLIBrKl7PL70XWxoJdWTSKcMaJAf6VhlKGoFqqaO275sVpVUFoVnMZn9WDUSc8q3fLDoyl8/wTljHQMs3xqhr0IloFhFdrv0M4FYAP9PtthvAfyipXZYDOMvM77o+2hBhdBMoSSrHxc3axplh5rQUGNJrUfAUqxCQ0XXulHuWLyg7H2c9brrR6rJGO4/LjwF4iopNddsA/B0zP0dEvwcAzPxdAD8EcBuAnwOYBPAfvRlueKgXAnEzE7Cy12LldwuCWzhJTJqcmrYlBpjVnix75hpWJaTdoNVFAsRu9TFzSHd3N4+M1EjaW5bOrc/butCNao8LwfLyyZcBAJ++5tMBj8Qf9JmuZhi2Tnz8kGHXLjcgAA+v74y9w0NEB5m522ibZIq6TKOeef/aJTV1XYxo9SllGGkVQ66h99rt9gU1a8HoFpFWWbiEGHQXaaZGh/4mSTisPS4ER6t56EBtI+96iVJbhg77UvXx3sEx9A6OVTXYbiUk5GIDu1632YXdSJjEaFqbUpKSmh9Cbv7BzQC8q4cedupdq1uGDtdt/uIFSoJw2cw2TEyqhvdtVHuVSsilCZx43W7W6BDduRAV6l2rj7x60urjAIoPgAQVC9i5hVrg8uKt/r6Na69SMeh1cFIZz+0aHWFKPgkbUfWu4orVtWrWi1ZDC49YlfR1g8r7Nq69St2q5RJbnHjdkp7vD5p3VVnPY/OTh0WnH1KSZJw1lyTC8YokOT/Wh7T7NgwVL71ADLoFQ6NZJEwuRqOLzyhB485lRW9AkoPco16Xe6cMjWaxYmCf/I084ks3XmPrfTeSmOqh3bdmD4+oiw4k5GKC5gUaTRetvO7KqWdc43ROcTs84qZ35cbfaMfndjj+3lbioZ6lAIqx9DwzkkT40o3XlN/X0M731qePNFTFtB5Kksr3bVybu8RW5dKsETFTrCSJ8J0vXG/rWG6qXqKKF2qdZs5r5XUxO6XgFxdUw7ILrfQ3CiP6+9cNyaOSIGz//PUALhXES5bkwVGSObacysUNr8vM2yswN32MqMfpnNDM4pPZQ7lR70p/XVhl5jr5G73wry8AcN7oQjBHv8hq9hB3glpgbH36CC5UdGXKM5evnSgY83rE0qDbNSJWXrwbipUwdKZxg2ZmO40+1Ow8lPt3Hykb5ZlKAiNvna6arus7xDupYc8oGhE7v+tDLz0EQAy6lxg9xBvBKJQTB3WLRiwNuh0jUs9guBFji0OcrtnZjt2Hmv6hMTk1bfpQHnnrNHYNn6hK9T4zqdYkr0zkVNxXksL1dGUcz4yyEzlseuxQ+fNCcNjpHaChNdJwEoePy6w5lioXOyvY9ZQSbnSVCVtnmkZoVlFiR8ppJEM0uxmzEzns1BlzKwooevJAsdKfU9QCo3/3EVHChACtH0CHYmy2OpQEjg+sweg3b8UDd1h3A9MTtVmzGbH00O14xna8eDcSe6KeHNTsOkBPVwYjb52uUjjcuSxTztaz43E1y0RORdeD9qpZmn2+snCa1sfyvsExMAHvKh+AiLBl6HCNckNwn5xasHx/aDSL/t1H6ha604jarNmKWBp0O2nzcYlve02z52loNIsnDmbL8s88M544WPRunziYbTomapdmZXBGxqEAlEv8MXM55GNk1CWz1T2srkm7pX01oqRusUNsZYv1kOJX9nB6nvSG6/zFaUPPOGlSTTKKqPQ2AEDh+UgS4Y1tt1Wdh3SHgnMXpqseCnKtNY7VNelkxhdVaaqVbLFlDTogXpNd7J4np96RED8P0S/Mrsnr+vY4qosexfMvBh1ivP3ADa1wFJlMvAoA6Cjc2NDnxVt3j0auQSfnPwx2pOUSi/R4lYIfhj9umIiL9Mspv2h7CgDQMdWYQc+pedz/qMgj3aARvbrdHBUAoS/l0RIGfevTR1wvlSl1WmpxK0W7Fckzt/z14wbauet1WIo3O5HDioF9hmse2r09oy0R+pK7sdShVzI0mjVVODTjUbpd8S8ObFq9yJH2V6im1a+fRjDKD+jpyuCe5QscHYeAch7EmUm1RtWUU/OmstcwzUwjb9DrJXxY3SDNSBSlTkstPV0ZXDazJSZ9npGdyGFh3x50bn1ekpfqYFUX/6GepUiZJCAZ0cxKYpikzpE26HYaHVgZ2GaSCeJaT7lZvCh72opoyUxi1M2pN0u+c9l8V7+vQ0mEvoFNpA26nbCHmYFNp5Sm4l7SnaiWVjU+V6n34yr1ftePqxZYQjAWWM2StYQ2O5h1VNIzo6SGCXMpj0jPj+2EPczKAPSvXdLUd7d6E2cjFUCrGp82nuvZsbMl49Qq15UTrDJGnVTWnNFGuDDNhnXxKzkzqWL73mOhvs8jrUO32+hA5IXuYpap16oJReeTLwEAZuVv8uT4BODu5QukTowOq4zRewfHmoqLWxF03oCVDt12yIWIkkQ0SkTPGGzbQETjRDRW+vc7zQzYLnbDHlqVtjcrGtIKjWMW6rI5c40dHyZ/iA+TP/Ts+Axg5/AJdD0oC6WVWFUz9XItK8xqJCchl68DeB3AR0y2DzLzHzQ/JPu0etgjKMy05jEpzRJazkyq6B0cw8hbp8VbL2FWzdSthhhmhFXNZsugE9F8AGsAfAvAfZ6OyCFRL08bdvThqoVXtraKJwzsHD6BncMnIlmHxC+cNMSwIkEwjK2HVc1mN+SyA8AfoVQx1IQ7ieg1InqciK4x2oGINhLRCBGNjI+POx2r4DNGstADb5wOelhCCSOZrnAJLdTaSCQwpSRxz/IFhgoYJUmhVbPV9dCJ6HYA7zPzQSK62WS3pwE8wswXiegrAP4GQE1dSmb+HoDvAcVF0YZHLbiOmWrFjSmrkiiWyrXZb0BwQE7No3dwrNT8OF9u8jCnQ8EDdyyp8d5bUSBgpoZJEqHAjKvTKaxcPBf7j47XXP9GdfBntbeF9pzVVbkQ0TYAXwYwDWAmijH0J5n5HpP9kwBOM/Nsq+OGoXyuUMQP1Uo6peDidCGWSpg8zgIAkrC85H0nmSB85/PXl41Pq/YAaPT3NivFSwDeHFjj/kBt0pTKhZk3M/N8Zl4I4IsA9umNORHNq3i5FsXFUyEimKlW7CZc2OFsTsW2dUtdPWZYSGJ26Iw5AOQLjPseHcOWocNYMbAPvYNjLVl/qNHevlHMBm84sYiIHgQwwsy7AfwhEa1F0Ys/DWCDO8MT/MBsxd7NjkIJIk+1wUFyLvkCAOCy/C0Bj6SWAqPcGs+MsCo23KQR8cSm1YuqeskCxfBhWOPngEODzswvAnix9PM3K97fDGCzmwMT/MOPsrdxaTdnRJgNuh3C7HH6iX59YeXiuahZUQ35BDPStVwEdzBK0GoGJUlIpxQQ7NfJEIKh1esPaQyNZrHpsUNViq6dwyeg5qsdETXP2Pr0EcsKr0ES6VougjvoE7Sa9aWn84yJnIqMNLwINaJjv0T/7iOGihYjzkyq5aqiYWtsIwZdAFAdY2y2N6h2W2QnciA0V2ta8I4odrz3CrPmFXYIU9ciCbkINbgZgmGEPuzYsiwMYcggqoRlYVkMulCDJvPqMOj4klKSSKcUR8eLu4f+0al+fHSqP+hhNIRkmxaZ0+HsmtbDQCgejmLQY0q91nx2YJ1vTQDuXJZB/9oljj34OHvpCcxEAjODHkbDtIIWvR4P3LEESrL6KtW/rkcYHo4SQ48ZQ6NZ9O8+UhUTrFy4AexVpzRKNmIA+4+Olyv9aceZnVJwfmq6RhGg/2xc+TC5BwBweT647MFmCUvIICjMKreaFfdKEhlKcYOOp4tBjxFGKc4aOTVfqvdxKf3eaoW+XjcofaLG0GgW9z96yLHefE6Hgo72NtNGJecvTje1YOUH55P/CCDaBl206ObJR2ZlA3oHxwyPk53IYcXAvkAURBJyiRH1immdmVRtp347TXvu6cqg0EDy0MSkatmo5Pbr5xl+zklHd6E+k1PTgcd/w4hV2QCrHIugwi/ioceIRqfN2ucqM+XSHQqUBFVpc62SUIZGs0iYTEOtuDqdsmxUYhbbnakkAVAsi30FwZlJFfdK8wxDzDz3etd6EOEXMegxwiqFP6UkMaMtYRi+SHcoNeGaM5NqOePzbE61jLdrn3VqzCsfEGY3jdlDamJSxcPrOy3j+PoHUiVaqEc7vtFeHUoCObUQ6/h/JQxg1/AJdF97RSg01WHHTuKc32sTkTLorVjL2Qlmbbe02tgAsOnxQzWLl2cmVWx+8rVyLW0NNc+YNaMNYw/cavm9jdRNTxLZrnhn1tndKI6vvz5G3jqNXcMnqoxySklW1QrX0r71RZi+ve6Thp+3w4y2BC5OW/WDCScMhCZJJuzYaXPn99pEZAy63oMMW8ptGLDTY1WvgNHQG3MNOx6GUy9ESRC2V9TptsLopjEL/Rh5+T1dGXRfe4XlObE6b5WfN/PG5qkDuHp2Cu9MVX92aDSL+x4di1xjj+xErhz7FQfKHH2bO31WdBB1cuo2uPAKpw0uzNLRM+mUpDA7wKxovxl2zq/Z30aTdlX2ZUynFPSvre2kY0VYZmYL+/aYbjtu0vBgaDRrOCsKO8kEIQHUrKHEvRlGM/h1nVo1uIiMh15PRifYw0mpXLsehpkX7dbNH5ZG4KYx08uexl+8/Dq+8elv1GzSxr316SPlgk5RIF9g6AMJQWusw04YrtPIGHSrWKpQH817sDLmlQuFTjwMO6GeOGD24EqmD+GZfz5qaNABY82+mYY57IgDdYmwzBwriYxBdxJLFaqxSjjS0C8UOiUM3onXmD24dhya4fg49R6uYUUcqCJhXdOLjEFvFS/QC8xUKJVdz+Vc2sPowbXjkPPjGLU3CzviQF3CrA9v0CGpyBh0oDW8wEaxmv6ZTZMLzIF2L29ltL+NWZ/VsNWRl2YY1YR1TU/yp2OANv2rbJ9VmXYcxe7lUSKlpJBSnJ9LK+PIKBrRMHDP8gU40LdKjHkFYb2nxKDHAKvpH2DcsEKmz+7x7N3P4tm7n23os2YGQJOLmskh/WTn8Aks7NuDrgefl3ovJcJ6T0Uq5CIYY6cyIiDrD2HEzmJ/WMIvZyZVbHq8uGCgJU616jUV1ntKDHoMsCPplPUH7/iz//dnAIA//bd/6viz9QzD0Gg2FMZcQ81zeeYXRpWHn4TxnopMpqhgjpEsUbL6/OPmH9wMAHhxw4uuHteO3DQICOZOhGRue08sMkUFc8I6/ROao5GiZwVjZL4AABEiSURBVH5wdToVWpVHqyMGPSaEcfonNEdYjaNVa7agVR6tjm2VCxEliWiUiJ4x2DaDiAaJ6OdE9CoRLXRzkILQioTROKZTCoBihyM9YVB5tDpOZItfB/C6ybbfBnCGmT8B4GEAf97swAQhKlzZcSWu7LjS9eMaSeM0lAQ57krvBkuuvhybnzxcU2gsnVJkzSYE2Aq5ENF8AGsAfAvAfQa7/AaA/tLPjwP4SyIiDmrFVRB85IkvPOHJcfX1trVyxFrWprbtnVJCmRWVJYyb4eU3Tht+16wZbWLMQ4DdGPoOAH8E4HKT7RkAJwGAmaeJ6CyAKwGcqtyJiDYC2AgACxYsaGS8gtBS1Fsb0bbV6xcwNJrFnzx1GOenmltkNXsmhDXe32rUDbkQ0e0A3mfmg81+GTN/j5m7mbl77ty5zR5OEELB5hc2Y/MLmwMdg1XmoiZ/bNaYWxHGeH8rYieGvgLAWiI6DuDvAawiop26fbIArgEAImoDMBvABy6OUxBCyytvv4JX3n4l0DH0dGWwbd1SZNIpEIqeuRbT9kP+mJ3IYcXAPikNEDB1Qy7MvBnAZgAgopsBfIOZ79HtthvAbwF4BcBdAPZJ/FwQ/MUsPONXOKQVs0XDRsPFuYjoQSJaW3r5fQBXEtHPUVw07XNjcIIgNI+f4ZDKonCC/zgy6Mz8IjPfXvr5m8y8u/TzBWb+PDN/gplvYOZ/9WKwgiA4x0z+qHhUa1UWSINDMkWFSBOGin/zPzLf1+9zillpCK/a4MkCaXCIQRciS1j6Ou5cp9cIhA+j+Pq9HjWqXrlYFGxBIQ0uhMhSr7GHxtBoFisG9uG6vj2ixKjAK0/6mUPvenJcoT5i0IXIYqfiX732fG7Q+1wvep/rde14fmEUW1eSBCXRXEmBiZxafyfBEyTkIkQWO409+ncfMfTiewfH0Ds4hpSSwLZ1nwRQHWNeuXgu9h8dtxWbH3vPm9CF15jF1ivfS5TKDThlaDQr0kX4v8YjDS6EyGLW2OPOZRnsPzruaMGvXq0TAnD38gV4qGdpzTavGlyEgUabbEiDFe8az1g1uJCQixBZjLIj71yWwRMHs47VG/UKVzGAXcMnWi7+rp1jrWyuXUSPbn+Nx00k5CJEGr16Y8XAPs/S3BnFmzTuXqdRmKB/7RJsffpIuWyuZuCt4uWtrkcPoquTGHQhVnhtRLITOVzXt6cqHtqODF47ebbm/ShiJAXd9NghgIoNojUuThfKsyGzB2ir69HtrPG4jYRchFjhhxHR1DK9g2NY2LcH/3L032Pm+a96pqLxE6MwgVrgKmMOFEMH+4+Om4ZjpHuRdQVMrxCDLsSKTasXwe8+Pvrwe5Tjx05mOO9M5NDTlcHYA7dix/pOw0qPrYxVBUyvkJCLECt6ujIYees0dg2fqNvFxy0+UP4bAOBK9Wvl96IaPzYLE5jtq6GtZWjx93sHx7B977FIh5/cwO/m7eKhC7HjoZ6leHh9J5Lkj6+uUhYqVYdYoho/Nkw2MuhfqiQJ5y9OV2Xf+pHEJVgjHroQS3q6Mp7VKqlHSkli5eK5WDGwL9CiYY1gJ9ko3aHg3IXpssJFM9wzlYSpTC8Kv3scEIMuxBYn4QM3mT9nZpX6I2qNH8zCBJX9SzX5okZOzZuqXaIafooiEnIRYktQVf/+5f3zvieU+IlTAx3V8FMUEYMuxJKh0SyeOOhP7LadP452/njd/eLiqZoZ6HRK8V2mJ1QjBl2IJX40Rta4Qt2IK9SNdfeLi6dqpq/uX7vEd5meUI3E0IVYEjZvOE6eqtnCqfa+GPDgEIMuxBI/F0RPKX8BALhK/Ybh9iRR7DxVv/XVgj3EoAuRY2g0W1Moqn/tkqrEFj/VLdN0ynJ7nlmMn+ALUg9dCCXFJJXXkFMLtj+jJAAQ1dQdubSdoNark9sA77X3AQB+aWrAdJ90SsHZnBopTboQTqzqoYuHLgSGvkzrysVz8cyhdxtuYVa0/eYG2wtjbhd9Eg4gsWbBfcSgC75RacC1bEPNyGYnctg5fCLgEfpDTs3j3sExbH36CCYm4+m1+916TSgiBl3wBX2dbX2moV8oCc2Td48ZhcWOP8O4dA7i5rUb1VSP0+8XZkSHLviCn7pwK9w25gAwZ3oD5kxvaOoYccokDaL1mlCkrkEnoplE9GMiOkRER4hoq8E+G4honIjGSv9+x5vhClElbLrwMBKXcxRE6zWhiJ2Qy0UAq5j5HBEpAH5ERM8y87Buv0Fm/gP3hyhEnS1Dh32rTR4E4+3fBgDMnfrjpo4Tl0zSIFqvCUXqGnQu6hrPlV4qpX9xvj8FF9kydNiTxc7zyZfwYfKHNe/PndqMJGbjXPIFnEu+ULP9o1P9SGAmPkzuwfnkP9Zs16SHZ9ueRC7x46pthBn42FRxgjrR9gguJA4BAKYSb6K9cF1Tv0+cMkk3rV5UFUMH4vX7hRlbi6JElARwEMAnAPwVM79qsNudRHQTgH8GcC8znzQ4zkYAGwFgwYIFDQ9aCD9BJPgERXvhOszK39zUMSpjzFFfOKxXGkDwDkeJRUSUBvAUgK8x888q3r8SwDlmvkhEXwGwnplXWR1LEovii17l4DbTNA4AaONgyuN6CQG4e/kCPNSzNOihCCHFKrHIkcqFmScA7AfwOd37HzDzxdLLvwawrJGBCvHAa0XLKeU7OKV8x7PjBwkD2DV8Qtq2CQ1hR+Uyt+SZg4hSAD4L4Khun3kVL9cCeN3NQQrRQtQMzcGASPyEhrATQ58H4G9KcfQEgEeZ+RkiehDACDPvBvCHRLQWwDSA0wA2eDVgIfwE1fotTshDUWgEOyqX1wB0Gbz/zYqfNwPY7O7QhKhipHIQnCESP6ERJPVfcB29yqG9LYGL0x6kaMYUAkTiJzSEGHTBE/QNELYMHcau4ROuJDB8ZPo3XThKeLl7+YLYSPykSJe/SC0XwRce6lmKNwfWIJ1SmjqOkiRsvOHzyMz8jEsjCxc71nfGRrKoyVezEzkwLhXpEgWPd4hBF3ylf+2SmgbDlVDp/0w6hR3rO7FjfWdV0+Htd12PL3+mHX//+9dWbUunFMzpUMr77VjfiTkdzh8emXSqoc+5AdXfJVJIkS7/kZCL4Cv6+PrslAIiWNYF17+++Qc3AwBe3PBi3en7vYNjtsI8SoKw/fPXo6crg+v69tj+fdxEkyvGJSQhRbr8Rwy64Dt+NRju6cpg5K3TdWP3lT1JgWBll3EydlKky3/EoAux5qGepei+9gpHC3ObVi+y7dm7TZyMnRTp8h8x6ELscToj0Dz7IFrixcnYSZEu/xGDLggGaJ597+BY0EOJNH6F14QiYtCFyLHlpi2+fE8QnnqcFkWFWrzW5YtBFyLHLR+/xbfv2n903LfvAuK1KCpU40fzbNGhC5Fj7L0xjL3nTyjEbwMbp0VRoRo/dPnioQuRo/e5XgBFHbrX+ClhlBou8cYPXb546IJgwabViywzW92kLRm3XFGhErPZl5uzMjHogmBBT1cG29YtLZcYSJJ3RlfNs6TFxxgj58BtXb6EXAShDpXSuy1Dhz1VvciiaHzxQ5cvBl0QHOC16kUWReON17p8MehC5Pj2v/t2YN/ttQe9cvFcT48vxBsx6ELk+PQ1nw7su71WvfitexfihSyKCpHj5ZMv4+WTLwfy3V6rXiSGLjSDeOhC5Pjj//vHAPzRoeupXNjywlOPQwxd2s4Fhxh0IbJsGTqMR149iTwzkkT40o3X+NK+TVvY0qdyOyEBIJkkqPlLRXrjUFrWj/R2wRwx6ELo0UsF32v/AABwfOrSe3nm8j5+GHXNC82peSSJkGcGEcA2i6gXAHykvQ2zZrTFypO1Sm+P+u8WBcSgC6HGqe575/AJzw263gvNMyOlJHHnsgwe+fFJ5Av2rPrZnIqxB271cqi+I23ngkUWRYVQE0STiXqYeaF7XnvX0Q0Vh3i5Hj/S2wVzxKALkeMKdSOuUDcG9v1m3uaZSRWqTe88DvFyI/xIbxfMqRtyIaKZAF4CMKO0/+PM/IBunxkA/hbAMgAfAFjPzMddH63gOwv79tS8d3xgTQAjuUQ7fzzQ729Gi06lz8chXm6EtJ0LFjsx9IsAVjHzOSJSAPyIiJ5l5uGKfX4bwBlm/gQRfRHAnwNY78F4BR8xMuba+0Ea9VyiWAs9Veg03D40mvXUgJg1P57RlsBETjX9XCadwoG+VZ6NKyxI27ngqGvQmZkBnCu9VEr/9PPK3wDQX/r5cQB/SURU+qwQA95r76t6ffMPtuMLS76A3/83v49JdRK37bqt5jMbOjdgQ+cGnJo8hbsevatm+1e7v4r1v7oeJ8+exJef+nLN9vs/dT+ABFR6Gx8of1l+fyrxJtoL1yE1ZWzQvVZUmHmhAExljBJ2EPzAlsqFiJIADgL4BIC/YuZXdbtkAJwEAGaeJqKzAK4EcEp3nI0ANgLAggULmhu50LK0F67DrPzNptv9aEhh5YVqSUeanDEjYQfBJ8iJE01EaQBPAfgaM/+s4v2fAfgcM79dev0GgBuZ+ZTxkYDu7m4eGRlpeOCC95iFXAD/4uhWYzAjSYQ3ttXOGAQhDhDRQWbuNtrmSOXCzBMA9gP4nG5TFsA1pS9rAzAbxcVRQfCdvET6hBalrkEnorklzxxElALwWQBHdbvtBvBbpZ/vArBP4ufRx8wLD1rlUo+MaJ6FFsVODH0egL8pxdETAB5l5meI6EEAI8y8G8D3AfxvIvo5gNMAvujZiAVfCbvxNkIWH4VWxY7K5TUAXQbvf7Pi5wsAPu/u0AQBuGf5AsfZorL4KLQqkikqhJqHepbinuX2FVFO9hWEuOFI5eImonIRGuHu//kKDrxxuuZ9P8vnCkKQWKlcpNqiECl2/e6ngh6CIIQWCbkIgiDEBDHogiAIMUEMuiAIQkwQgy4IghATxKALgiDEhMBki0Q0DuCtQL7cmKugqw4pVCHnxxo5P9bI+bHGyfm5lpnnGm0IzKCHDSIaMdN2CnJ+6iHnxxo5P9a4dX4k5CIIghATxKALgiDEBDHol/he0AMIOXJ+rJHzY42cH2tcOT8SQxcEQYgJ4qELgiDEBDHogiAIMaHlDToR/S8ier/U6FrQQUTXENF+IvonIjpCRF8PekxhgohmEtGPiehQ6fxsDXpMYYSIkkQ0SkTPBD2WsEFEx4noMBGNEVFTNcVbPoZORDcBOAfgb5n5V4MeT9ggonkA5jHzT4nocgAHAfQw8z8FPLRQQEQEYBYznyMiBcCPAHydmYcDHlqoIKL7AHQD+Agz3x70eMIEER0H0M3MTSdetbyHzswvodgHVTCAmd9l5p+Wfv4QwOsApMdbCS5yrvRSKf1rbS9JBxHNB7AGwF8HPZa40/IGXbAPES1Esb/sq8GOJFyUwgljAN4H8A/MLOenmh0A/ghAIeiBhBQG8DwRHSSijc0cSAy6YAsiugzAEwB6mfkXQY8nTDBznpk7AcwHcAMRSeiuBBHdDuB9Zj4Y9FhCzGeY+dcA/DqA/1QKAzeEGHShLqXY8BMAdjHzk0GPJ6ww8wSA/QA+F/RYQsQKAGtLceK/B7CKiHYGO6RwwczZ0v/vA3gKwA2NHksMumBJadHv+wBeZ+b/HPR4wgYRzSWidOnnFIDPAjga7KjCAzNvZub5zLwQwBcB7GPmewIeVmggolklsQGIaBaAWwE0rLhreYNORI8AeAXAIiJ6m4h+O+gxhYwVAL6Momc1Vvp3W9CDChHzAOwnotcA/ATFGLpI8wS7fAzAj4joEIAfA9jDzM81erCWly0KgiDEhZb30AVBEOKCGHRBEISYIAZdEAQhJohBFwRBiAli0AVBEGKCGHRBEISYIAZdEAQhJvx/TvdOe/qODWIAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}