Planar Crane with SymPy.ipynb

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   "source": [
    "<h1 style=\"text-align:center\">Planar Crane Example</h1>\n",
    "<h3 style=\"text-align:center\">MCHE 513: Intermediate Dynamics</h3> \n",
    "<p style=\"text-align:center\">Dr. Joshua Vaughan <br>\n",
    "<a href=\"mailto:joshua.vaughan@louisiana.edu\">joshua.vaughan@louisiana.edu</a><br>\n",
    "<a href=\"http://www.ucs.louisiana.edu/~jev9637/\">http://www.ucs.louisiana.edu/~jev9637/</a></p>"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "In this example, we'll determine the equations of motion for the planar crane model shown in Figure 1. The system consists of a point mass, $m$, connected to the crane trolley, which has mass $m_t$, by an ideal pin via a massless rod of length $l$. The rotation of the suspension cable is represented by $\\theta$. There is a force, $f$, acting on the crane trolley.\n",
    "\n",
    "<p style=\"text-align:center\">\n",
    "\t<img src=\"http://shared.crawlab.org/crane_forwardAngle.png\" alt=\"Planar Crane\" width=35%><br>\n",
    "    <strong>Figure 1: Planar Crane Model</strong>\n",
    "</p>"
   ]
  },
  {
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   "execution_count": 1,
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   "source": [
    "# Import the SymPy Module\n",
    "import sympy\n",
    "\n",
    "# Import the necessary sub-modules and methods for dynamics\n",
    "from sympy.physics.mechanics import dynamicsymbols\n",
    "from sympy.physics.mechanics import LagrangesMethod, Lagrangian\n",
    "from sympy.physics.mechanics import Particle, Point, ReferenceFrame\n",
    "\n",
    "# initiate better printing of SymPy results\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   "outputs": [],
   "source": [
    "# Define the genearlized coordinates and other dynamic symbols\n",
    "x, theta, f = dynamicsymbols('x theta f')\n",
    "\n",
    "# Also define the first derivatives\n",
    "x_dot, theta_dot = dynamicsymbols('x theta', 1)\n",
    "\n",
    "# Define the symbols for the other paramters\n",
    "m, mt, g, l, t = sympy.symbols('m, m_t, g, l, t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "outputs": [
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      "text/latex": [
       "$$g l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\frac{l^{2} m}{2} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2} + l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\theta{\\left (t \\right )} \\frac{d}{d t} x{\\left (t \\right )} + \\frac{m}{2} \\left(\\frac{d}{d t} x{\\left (t \\right )}\\right)^{2} + \\frac{m_{t}}{2} \\left(\\frac{d}{d t} x{\\left (t \\right )}\\right)^{2}$$"
      ],
      "text/plain": [
       "                                 2                                            \n",
       "                   2   ⎛d       ⎞                                        ⎛d   \n",
       "                  l ⋅m⋅⎜──(θ(t))⎟                                      m⋅⎜──(x\n",
       "                       ⎝dt      ⎠                  d        d            ⎝dt  \n",
       "g⋅l⋅m⋅cos(θ(t)) + ──────────────── + l⋅m⋅cos(θ(t))⋅──(θ(t))⋅──(x(t)) + ───────\n",
       "                         2                         dt       dt               2\n",
       "\n",
       "     2                 2\n",
       "    ⎞        ⎛d       ⎞ \n",
       "(t))⎟    m_t⋅⎜──(x(t))⎟ \n",
       "    ⎠        ⎝dt      ⎠ \n",
       "────── + ───────────────\n",
       "                2       "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define the Newtonian reference frame\n",
    "N = ReferenceFrame('N')\n",
    "\n",
    "# Define a body-fixed frame along the pendulum, with y aligned from m to the pin\n",
    "A = N.orientnew('A', 'Axis', [theta, N.z])\n",
    "\n",
    "# Define the trolley point and its velocity\n",
    "T = Point('T')\n",
    "T.set_vel(N, x_dot * N.x)\n",
    "\n",
    "# Treat the trolley as a particle\n",
    "mtr = Particle('mtr', T, mt)\n",
    "\n",
    "# Define the payload point and set its velocity\n",
    "P = Point('P')\n",
    "P.set_vel(N, x_dot * N.x + l * theta_dot * A.x)\n",
    "\n",
    "# The payload is a particle (a point mass)\n",
    "mp = Particle('mp', P, m)\n",
    "\n",
    "# Define the potential energy of the payload - just gravity here\n",
    "mp.set_potential_energy(-m * g * l * sympy.cos(theta))\n",
    "\n",
    "# Set up the force list - each item follows the form:\n",
    "#    (the location where the force is applied, its magnitude and direction)\n",
    "forces = [(T, f * N.x)]\n",
    "\n",
    "# Form the Lagrangian, then simplify and print\n",
    "L = Lagrangian(N, mtr, mp)\n",
    "L.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# This creates a LagrangesMethod class instance that will allow us to form the equations of motion, etc\n",
    "LM = LagrangesMethod(L, [x, theta], forcelist = forces, frame = N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   "outputs": [
    {
     "data": {
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DadZw9gJOYXnGYR6yujV27GKphGxuOInFFjtIQO4T9URdbwqc2fHcJG1tMZ0O3JOExEHE\nq2oENqA0x43H2RNOIf+soyh7XVIJ2dxwEvvqghdQFvHWpCO5L4gvSd+bAmORxIWxYuOnT2lqIE9L\nQuWDVBVYabgH3qsNcfZcB6wab+w6CuMLJzGGShhP7nM9djfuTYHrwX19JcitelINyg0IZLMFxzP0\nZNtR9mp9zp7rgPXjjV1vYXyz2aMpi+5bkzCE6yjhuCcFLj/PPPiVm6GNqcE6qKvtwpUMIzh7joOV\nLENzYXzGjqUSpoMIIQanPSlw9v7N1UjvjlDtYXvMOJO0oxcaRnD2lKdHQpodTsP4jC34Cch9HSAi\np3tS4EjPYaG8KWJhp1dGrmEkZ886cFU7jX18+8ooDbmvE5Yzf+AK7IAmGKbm7HWNsZ/kvk44B3uB\nO3D2OqXXxfx+kvs6IXCBX//PQTtVKfX84/yfg558gP5z0NRRvoYND1mzpvk/TfnHiD/MQ/QAAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2} + l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )} + m \\frac{d^{2}}{d t^{2}}  x{\\left (t \\right )} + m_{t} \\frac{d^{2}}{d t^{2}}  x{\\left (t \\right )} - f{\\left (t \\right )}\\\\g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + l^{2} m \\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )} + l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d^{2}}{d t^{2}}  x{\\left (t \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                          2                   2             2               2\n",
       "⎢                ⎛d       ⎞                   d             d               d \n",
       "⎢- l⋅m⋅sin(θ(t))⋅⎜──(θ(t))⎟  + l⋅m⋅cos(θ(t))⋅───(θ(t)) + m⋅───(x(t)) + m_t⋅───\n",
       "⎢                ⎝dt      ⎠                    2             2               2\n",
       "⎢                                            dt            dt              dt \n",
       "⎢                                                                             \n",
       "⎢                                         2                         2         \n",
       "⎢                                   2    d                         d          \n",
       "⎢                g⋅l⋅m⋅sin(θ(t)) + l ⋅m⋅───(θ(t)) + l⋅m⋅cos(θ(t))⋅───(x(t))   \n",
       "⎢                                         2                         2         \n",
       "⎣                                       dt                        dt          \n",
       "\n",
       "             ⎤\n",
       "             ⎥\n",
       "(x(t)) - f(t)⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎥\n",
       "             ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LM.form_lagranges_equations()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The <tt>LagrangesMethod</tt> class gives us lots of information about the system. For example, we can output the mass/inertia matrix and the forcing terms. Note that the forcing terms include what might be conservative forces and would therefore normally appear in a stiffness matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPwAAAAyBAMAAACKZ6uQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}m + m_{t} & l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\\\l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & l^{2} m\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   m + m_t     l⋅m⋅cos(θ(t))⎤\n",
       "⎢                            ⎥\n",
       "⎢                    2       ⎥\n",
       "⎣l⋅m⋅cos(θ(t))      l ⋅m     ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Output the inertia/mass matrix of the system\n",
    "LM.mass_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQoAAAA2BAMAAADNKsqYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2} + f{\\left (t \\right )}\\\\- g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                        2       ⎤\n",
       "⎢              ⎛d       ⎞        ⎥\n",
       "⎢l⋅m⋅sin(θ(t))⋅⎜──(θ(t))⎟  + f(t)⎥\n",
       "⎢              ⎝dt      ⎠        ⎥\n",
       "⎢                                ⎥\n",
       "⎣        -g⋅l⋅m⋅sin(θ(t))        ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Output the forcing terms of the system\n",
    "LM.forcing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use builtin functions to write the sytsem as a set of first order ODEs, suitable for simluation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cnrtDD/sjLAoQAs2GC004sKBHat2oMombTPzascIWvUqkjgoJGjxU9k/BMgNPX3260ZDd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\n8lNdCs7Bnt9ctir+ON+EVSlfTJRD8P/+CksCe1LskgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{d}{d t} x{\\left (t \\right )}\\\\\\frac{d}{d t} \\theta{\\left (t \\right )}\\\\\\frac{1}{m \\sin^{2}{\\left (\\theta{\\left (t \\right )} \\right )} + m_{t}} \\left(\\frac{g m}{2} \\sin{\\left (2 \\theta{\\left (t \\right )} \\right )} + l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2} + f{\\left (t \\right )}\\right)\\\\- \\frac{1}{l \\left(m \\sin^{2}{\\left (\\theta{\\left (t \\right )} \\right )} + m_{t}\\right)} \\left(g \\left(m + m_{t}\\right) \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + \\left(l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2} + f{\\left (t \\right )}\\right) \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                d                                       ⎤\n",
       "⎢                                ──(x(t))                                ⎥\n",
       "⎢                                dt                                      ⎥\n",
       "⎢                                                                        ⎥\n",
       "⎢                                d                                       ⎥\n",
       "⎢                                ──(θ(t))                                ⎥\n",
       "⎢                                dt                                      ⎥\n",
       "⎢                                                                        ⎥\n",
       "⎢                                                     2                  ⎥\n",
       "⎢           g⋅m⋅sin(2⋅θ(t))                 ⎛d       ⎞                   ⎥\n",
       "⎢           ─────────────── + l⋅m⋅sin(θ(t))⋅⎜──(θ(t))⎟  + f(t)           ⎥\n",
       "⎢                  2                        ⎝dt      ⎠                   ⎥\n",
       "⎢           ──────────────────────────────────────────────────           ⎥\n",
       "⎢                                2                                       ⎥\n",
       "⎢                           m⋅sin (θ(t)) + m_t                           ⎥\n",
       "⎢                                                                        ⎥\n",
       "⎢ ⎛                        ⎛                        2       ⎞          ⎞ ⎥\n",
       "⎢ ⎜                        ⎜              ⎛d       ⎞        ⎟          ⎟ ⎥\n",
       "⎢-⎜g⋅(m + m_t)⋅sin(θ(t)) + ⎜l⋅m⋅sin(θ(t))⋅⎜──(θ(t))⎟  + f(t)⎟⋅cos(θ(t))⎟ ⎥\n",
       "⎢ ⎝                        ⎝              ⎝dt      ⎠        ⎠          ⎠ ⎥\n",
       "⎢────────────────────────────────────────────────────────────────────────⎥\n",
       "⎢                           ⎛     2            ⎞                         ⎥\n",
       "⎣                         l⋅⎝m⋅sin (θ(t)) + m_t⎠                         ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make the call to set up in state-space-ish form q_dot = f(q, t)\n",
    "lrhs = LM.rhs()\n",
    "\n",
    "# Simplify the results\n",
    "lrhs.simplify()\n",
    "\n",
    "# Output the result\n",
    "lrhs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also linearize these equations with builtin SymPy methods. Let's do so about the operating point:\n",
    "\n",
    "$$ x = \\dot{x} = \\theta = \\dot{\\theta} = 0 $$\n",
    "\n",
    "The resulting equations returned are a system of first order ODEs in state-space form:\n",
    "\n",
    "$$ \\dot{x} = Ax + Bu $$\n",
    "\n",
    "See the [SymPy Documentation](http://docs.sympy.org/0.7.6/modules/physics/mechanics/linearize.html#linearizing-lagrange-s-equations) for much more information."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Define the point to linearize around\n",
    "operating_point = {x: 0.0, x_dot: 0.0, theta: 0.0, theta_dot: 0.0}\n",
    "\n",
    "# Make the call to the linearizer\n",
    "A, B, inp_vec = LM.linearize([x, theta], \n",
    "                             [x_dot, theta_dot],\n",
    "                             op_point = operating_point, \n",
    "                             A_and_B = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAACCCAMAAADG3HUqAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 1 & 0\\\\0 & 0 & 0 & 1\\\\0 & \\frac{g l^{2} m^{2}}{\\left(m + m_{t}\\right) \\left(- \\frac{l^{2} m^{2}}{m + m_{t}} + l^{2} m\\right)} & 0 & 0\\\\0 & - \\frac{g l m}{- \\frac{l^{2} m^{2}}{m + m_{t}} + l^{2} m} & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0               0                1  0⎤\n",
       "⎢                                     ⎥\n",
       "⎢0               0                0  1⎥\n",
       "⎢                                     ⎥\n",
       "⎢                2  2                 ⎥\n",
       "⎢             g⋅l ⋅m                  ⎥\n",
       "⎢0  ────────────────────────────  0  0⎥\n",
       "⎢             ⎛    2  2        ⎞      ⎥\n",
       "⎢             ⎜   l ⋅m      2  ⎟      ⎥\n",
       "⎢   (m + m_t)⋅⎜- ─────── + l ⋅m⎟      ⎥\n",
       "⎢             ⎝  m + m_t       ⎠      ⎥\n",
       "⎢                                     ⎥\n",
       "⎢             -g⋅l⋅m                  ⎥\n",
       "⎢0        ────────────────        0  0⎥\n",
       "⎢             2  2                    ⎥\n",
       "⎢            l ⋅m      2              ⎥\n",
       "⎢         - ─────── + l ⋅m            ⎥\n",
       "⎣           m + m_t                   ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUkAAACNCAMAAAANWniMAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRS\nTlMAMquZdlQQQO0wRIlmzd0i77t8bBwggJIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAsvSURBVHgB\n7V2LuqMoDPZ+ZqvVdtf3f9blIkpCQKBg9Yx+30w1QAK/XAJJPEU5i6sq7isOgVECWBTlXDfsauPY\nHFGqq/uhL4+QFCVj4PBVM0fyvJWULXuyt9yNU1Qzjyo0XAHJ4cXh6J9HgRIl5xJIvsUU3s5dVBMP\nKnQJJGeB5GM+8UxeFFdAsptr3q8ec3NQ94oScwUkp7nnbSvlT1QzDyh0DSRFn7yR/Lg/3KP7YwgV\nA7nitPeKowCJ/n2/edHh1oKiEVQFpWZe35q5AiT+d+S7xde597RXWLsZij07ITg3kJfQzOP78pEl\nr9Enj0QkVtaNZCxyuNyNJEYk9vlGMhY5XO5GEiMS+3wjGYscLncjiRGJfb6RjEUOl7uRxIjEPt9I\nxiKHy10DyZN7DghQr4Hk7TmAR0Dk8+05EAmcUez2HDAgiSTcngORwOFit20RIxL7fHsOxCKHy03S\nm+X2HMDABD/fozsYMluB23PAhkwo/TKeA117ahdP5n0hfHpP7jkwMZv8+f3Mb8+B0GFsy397DtiQ\n+Y30a5wFXQH5G8lUb+lG8kYyFQKp+Nx98kYyFQKp+GTok/0jVeXOyqejArjTI9mL4JmzgpCmXg/C\n5T05kj31vtLU/0RcBhNKjuTP/JOsku3r5KchiVr6FgGAOrN/08Z3d69fP0lK9LoXDuxNPLrff8Ek\nKaFsxUmf1inTInnyYHat3Z/fjmh8p0Vy/Gu6ZFG0M/wuR1IkB8T88xe/cij7/g1rLpIs5LVYzhvU\nbZIiOYpIzSy1Z5wfo8nZQjYzZqCgiNSUSJZzvoWb6W8dEUtrIWfAzWTZwa9JpESyxsuZKf0TSkv0\nScbPQv5EkmfZJ1DP/ZAsQRmboFeW7U33ln2xexLzJOupNNlWx5T0BiwLPkiWdUV3B1itch4gIdGT\nnB+7itw8WciJRLvZTKDBPkh6jiD4ityVCEkVRyJdPXVIgeM8LOQQ9h/kBYOQI9k15OvWRHjNRe9M\n0+STd3X+OT1iYFjIWtVz3r71Gk3sa0b7ngNeSL4y6UBwNqqHsmnqomne34+c71k3XK90o3vxKFs5\np7opQVcfHnxyerFJs8myvgXVetDVvnRIllC9CqqSK3Ojd/Wpa/iI4i2oiWnTxSdJGjw0BE1Oh2Su\nz/m84dfUKj4d8W7KfYVyXy2Q3VV1BXYHYBimQ3LI9D3QVwkmRP6FFt5Np7kA9CygQiSZiAYgWUjH\nTinaE0kwV9GVzqUEzYXeMRiARcH75fAu8g/vXSS1mccHyakeZ15599WDNdadNyS1BrbKB19n+LZm\nqvo93S1ECp13D8mXtvfzQZKWgqk17Pg4+ZLPu0hqCmU6JPm3k3/btYfkqM16CEn5Ye6Q/1fwIJIh\nLE6Vd2lP/+TX+BI/m+aKVhwHkisu4Td/4+ges4zuHCuOa1EhDQ8k0aNXWOxPe6MbrzhpPAea9Ppk\n7dIYmQJi2iNIogeSpabOaNl3kdSK+XgObNOEJsS4BZtQIzWGUDvPO0nDA0n0kd2SvXIPSRkGKPmj\nFYcUqgFPpktiq2/nHfm8kyz9hJWfRokxeURFEveFvqnNJ0Kyrl7zs9reb6d/zRojORCnVgpJKm2r\nItjOb+T4u6d9bMsTNdLwQBK3SlgnUtLAgpDc2Mg70GQDSeLUakWSSNOY611dkknDi1aCuN3Qa5VY\nIpdQ40jDA0nUODCm5uwq0p+EYbRztwAc2iAkyVOrpUlG2gQn0FHbOom6xeyL27U92ijSkBC3E6+R\nMjwMdTsM9aMVf/VFEXEJ/ixMa/aJtMXVp3hAGlBXEJLydGA7taq4ZirUU77vRidaA9yL12iTU7lf\nKKzU+lSrUo7N58DfkTI8tA2DgHmOtTpxZaffLJ3RMpFS5nS9NHEPNiMGksSp1TrMYFo/wsUV+SSU\nti4JezKuoPI8fmgbMZynWjsuT2Ef8O2YhlwDolZklSe9ja0T6WhjoPGCt8DegpGkTq0UkjhNU/C5\nhBJ+ud36hWLFDtZqfVrs170j29jpCy2rBtdhxkInruyKYmUkTGv2ibSyvXqNF7idgJUAI0mdWqmq\noLQOTyzAaMlaxi5quVfsqDRWpJFahsu40PdqCmD5eSduWHeqGo3IZatLySv4sZ+aSAnhrneneIFf\neCKLkQRZl4e1KjCRdYRNteJJYKKcJJLEcq/YWYxbrRyNyOgARQc9KXly9mfHJbxmhHA+3wZdwEjr\n9RVuVRUkZsK94KHr5g9eyljuOYuFnZG2qAKiIDvNhW9pkR1ybLQUUfKEzULRDOEsYRDvXuXY/yU8\nrPY8ByyjxpSlexSKqQAv9z6qwKIiL9uYon2rK+KIHMiDvRzpIawxhr+z2UBAQSdsfp4DgIP9QV+9\nl70eXO5F0bWLwzSlCiyv4AlVLKvQ/WVCyYOmNSicsyciQ6xSeQJcFrxGt5MfSHxt5wBcMREWQNkv\nNQOWaplFFRDKIpsD9iESojeJoCbag5IHTGtYOMsPDOtaecttVu9o5nu9DcGR36LlXlRKtQylKVVg\n0QvrfYgEt/1sSh4wrSHhnJWvRCGWTeRIss/avRT1+dHYW/+mqGoZ4qdUgWUFtexEwN6Qc1jaA+lg\nH2uRh8QzaDznE1kQemMwWmIkJ+2419YCC31RBdplze7Q3nNpONgb1lVVjewfU8kBvQD7WIs8A8kg\nK7MZwpUYSd1XWWGCq+xUBbp1zFiizfDeUOXX6WrxkpKd8rbKlUFKkBnClRrJol63uUXr2YatNWza\nXydaiy8aay/YGyokAT0IlEW8a1Ol11DcswUUX8mRFAdGWErUMxlLauwNFyQBXS1eQWJJcRYOh0Ql\nOw5lLNWykRsvPUj1SZ2LWrx02t59E7DekKcg6fvkXpX9072iGrZD9o2xsY/dkmx3Uv21pfrQz4zk\nFHqk4NNgS564Y2md2ZmRLMpt9VJROXrdE947Let+cjiSP3/+8ct8eK51IbfasRJVSRMUy/G/Pz6x\nD7HcE5Y7/YdKTj269RdBn1fqOb58fxkk+V6OMBl8GT5N/FWQFKYDwmSgNeXLt1dBUoQ7fDkYx/2q\nroKkNB2YJgN3645MvQqS0nRgmgyOxMot6ypICtMBYTJwt+7I1KsgKUwHhMngSKzcsq6CpLsVZ0i9\nkUz1FhIg2T3H9ZJ+uCGeEqfK+wmqCZD8RPwvKntiJL8djBP4ls+LpPPIkBmkTHdxkuiBhz3IwqPw\nmuW0SEJ/4bW+yw07TTe9mUkiLkk9c2Plx9dZkbT3k8OCcQKxPSuSWYJxrNiQwTjW3HRCPiQj3AY2\nO2GeYBxrWFNBBePQgFmp2ZD0MlajauUOxrEvSRHBOKjuqT2sVvZxVs/MwTj2JclcvtaW+N5k6pNn\nDcbRPcAAROHBOKA4e8iE5EmDcexfqwwOxsFA5kJSOItRBizl40WlscrlDcZhsTvCsE0IDw7GOQjJ\nkwbjrGFNZpRQcDDOQUjmCcYxKu9BUGNAuOMvQaMpgnFM0XnmSXG4nTwYJ+QAjrU0azDOUUj+/mCc\no5D89cE4JpDZ1m6+RlIGLDVvoTTlz3yVYJzjkPxmMA5opXpzgEg8hAXjEAxy9ck1gBXLtLQsZTBO\nVFhTUDAObpR4zrN2s0gPMqiYBTKStViISYJxYFiTU95Wl7BgnK2cdpcLye8F48CwJq2prtugYBya\nUTYkaXHeVDI6BgTdcFZUMI78bIS3IJmRFBfGQyIpdN7NPT6MRZbc8cE4Sg0IqVZIMA7Bl++d2MVs\nS424gj/yQvBMR4oOxokIa/o0GGeQAKZrfFJO0cE4EWFNccfSSZubk5kWjJNTDOPttKxnln0I+wQx\nMn71TCbof2gRnCmB8ydVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\\\frac{1}{m + m_{t}} \\left(\\frac{l^{2} m^{2}}{\\left(m + m_{t}\\right) \\left(- \\frac{l^{2} m^{2}}{m + m_{t}} + l^{2} m\\right)} + 1\\right)\\\\- \\frac{l m}{\\left(m + m_{t}\\right) \\left(- \\frac{l^{2} m^{2}}{m + m_{t}} + l^{2} m\\right)}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡               0                ⎤\n",
       "⎢                                ⎥\n",
       "⎢               0                ⎥\n",
       "⎢                                ⎥\n",
       "⎢            2  2                ⎥\n",
       "⎢           l ⋅m                 ⎥\n",
       "⎢──────────────────────────── + 1⎥\n",
       "⎢          ⎛    2  2        ⎞    ⎥\n",
       "⎢          ⎜   l ⋅m      2  ⎟    ⎥\n",
       "⎢(m + m_t)⋅⎜- ─────── + l ⋅m⎟    ⎥\n",
       "⎢          ⎝  m + m_t       ⎠    ⎥\n",
       "⎢────────────────────────────────⎥\n",
       "⎢            m + m_t             ⎥\n",
       "⎢                                ⎥\n",
       "⎢             -l⋅m               ⎥\n",
       "⎢  ────────────────────────────  ⎥\n",
       "⎢            ⎛    2  2        ⎞  ⎥\n",
       "⎢            ⎜   l ⋅m      2  ⎟  ⎥\n",
       "⎢  (m + m_t)⋅⎜- ─────── + l ⋅m⎟  ⎥\n",
       "⎣            ⎝  m + m_t       ⎠  ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given these two matrices, the system of equations match those that we'd expect:\n",
    "\n",
    "$$ \\begin{bmatrix}\\dot{w}_1 \\\\ \\dot{w}_2 \\\\ \\dot{w}_3 \\\\ \\dot{w}_4\\end{bmatrix} = A \\begin{bmatrix}w_1 \\\\ w_2 \\\\ w_3 \\\\ w_4 \\end{bmatrix} + B f(t)$$\n",
    "\n",
    "where:\n",
    "\n",
    "$$ \\mathbf{w} = \\begin{bmatrix}w_1 \\\\ w_2 \\\\ w_3 \\\\ w_4 \\end{bmatrix} = \\begin{bmatrix}x \\\\ \\theta \\\\ \\dot{x} \\\\ \\dot{\\theta}\\end{bmatrix} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation\n",
    "We can pass these equations of motion to numerical solver for simluation. To do so, we need to import [NumPy](http://numpy.org) and the [SciPy](http://www.scipy.org) ode solver, ```ode```. We'll also import [matplotlib](http://www.scipy.org) to enable plotting of the results.\n",
    "\n",
    "For a system as simple as this one, we could easily set up the necessary components for the numerical simulation manually. However, here will automate as much as possible. Following a similar procedure on more complicated systems would be necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# import NumPy with namespace np\n",
    "import numpy as np\n",
    "\n",
    "# import the ode ODE solver\n",
    "from scipy.integrate import ode\n",
    "\n",
    "# import the plotting functions from matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# set up the notebook to display the plots inline\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's define the forcing function, so that it can be substituted into the generated equations of motion. Here, we'll use a simple bang-bang input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# define the forcing function\n",
    "def force(t):\n",
    "    # set the maximum force output\n",
    "    fmax = 10.0\n",
    "    \n",
    "    # return a bang-bang command\n",
    "    return fmax * ((t > 0.25) - 2 * (t >= 1.25) + (t >= 2.25))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the input to make sure it matches the bang-bang profile we precribed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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W5AFJ9yUdKR9albR/xsPAD6rYDHZ4kgYxDzzTR+9EbNQnPmoUHzXKa1qP2PWxP5+WtLbN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\nSyG0yiHaWrIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110275a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set the plot size - 3x2 aspect ratio is best\n",
    "fig = plt.figure(figsize=(6, 4))\n",
    "ax = plt.gca()\n",
    "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n",
    "\n",
    "# Change the axis units to serif\n",
    "plt.setp(ax.get_ymajorticklabels(), family='serif', fontsize=18)\n",
    "plt.setp(ax.get_xmajorticklabels(), family='serif', fontsize=18)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "\n",
    "ax.xaxis.set_ticks_position('bottom')\n",
    "ax.yaxis.set_ticks_position('left')\n",
    "\n",
    "# Turn on the plot grid and set appropriate linestyle and color\n",
    "ax.grid(True,linestyle=':', color='0.75')\n",
    "ax.set_axisbelow(True)\n",
    "\n",
    "# Define the X and Y axis labels\n",
    "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n",
    "plt.ylabel('Force (N)', family='serif', fontsize=22, weight='bold', labelpad=10)\n",
    "\n",
    "# Plot the force over 10 seconds\n",
    "force_plot_time = np.linspace(0.0, 10.0, 1001)\n",
    "plt.plot(force_plot_time, force(force_plot_time), linewidth=2, linestyle='-', label=r'Force')\n",
    "\n",
    "# uncomment below and set limits if needed\n",
    "# plt.xlim(0, 5)\n",
    "# plt.ylim(0, 10)\n",
    "\n",
    "# Create the legend, then fix the fontsize\n",
    "# leg = plt.legend(loc='upper right', ncol = 1, fancybox=True)\n",
    "# ltext  = leg.get_texts()\n",
    "# plt.setp(ltext, family='serif', fontsize=20)\n",
    "\n",
    "# Adjust the page layout filling the page using the new tight_layout command\n",
    "plt.tight_layout(pad=0.5)\n",
    "\n",
    "# Uncomment to save the figure as a high-res pdf in the current folder\n",
    "# It's saved at the original 6x4 size\n",
    "# plt.savefig('Bang_Bang_Force.pdf')\n",
    "\n",
    "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this plot, you can see why this type of command is called bang-bang. It \"bangs\" on, then \"bangs\" negative, then is zero for all other time.\n",
    "\n",
    "Now, let's continue setting up the simluation by creating a version of the generated equations of motion that we can use in a numerical ODE solver."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Define the states and state vector\n",
    "w1, w2, w3, w4 = sympy.symbols('w1 w2 w3 w4', cls=sympy.Function)\n",
    "w = [w1(t), w2(t), w3(t), w4(t)]\n",
    "\n",
    "# Set up the state definitions and parameter substitution\n",
    "sub_params = {x : w1(t),\n",
    "              theta : w2(t), \n",
    "              x_dot : w3(t),\n",
    "              theta_dot : w4(t), \n",
    "              mt : 10.0,\n",
    "              m : 1.0,\n",
    "              g : 9.81, \n",
    "              l : 2.0, \n",
    "              f: force(t)}\n",
    "\n",
    "# set this parameter to enable array output from sympy.lambdify\n",
    "mat2array = [{'ImmutableMatrix': np.array}, 'numpy']\n",
    "\n",
    "# Create a function from the equations of motion\n",
    "# Here, we substitude the states and parameters as appropriate prior to the lamdification\n",
    "eq_of_motion = sympy.lambdify((t, w), \n",
    "                              lrhs.subs(sub_params), \n",
    "                              modules = mat2array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Set up the initial conditions for the solver\n",
    "x_init = 0.0                   # Initial trolley position (m)\n",
    "x_dot_init = 0.0               # Initial trolley velocity (m/s)\n",
    "theta_init = 0.0 * np.pi/180   # Initial angle (rad)\n",
    "theta_dot_init = 0.0           # Initial angular velocity (rad/s)\n",
    "\n",
    "# Pack the initial conditions into an array\n",
    "x0 = [x_init, theta_init, x_dot_init, theta_dot_init]\n",
    "\n",
    "# Create the time samples for the output of the ODE solver\n",
    "sim_time = np.linspace(0.0, 10.0, 1001) # 0-10s with 1001 points in between"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Set up the initial point for the ode solver\n",
    "r = ode(eq_of_motion).set_initial_value(x0, sim_time[0])\n",
    " \n",
    "# define the sample time\n",
    "dt = sim_time[1] - sim_time[0]   \n",
    "\n",
    "# pre-populate the response array with zeros\n",
    "response = np.zeros((len(sim_time), len(x0)))\n",
    "\n",
    "# Set the initial index to 0\n",
    "index = 0\n",
    "\n",
    "# Now, numerically integrate the ODE while:\n",
    "#   1. the last step was successful\n",
    "#   2. the current time is less than the desired simluation end time\n",
    "while r.successful() and r.t < sim_time[-1]:\n",
    "    response[index, :] = r.y\n",
    "    r.integrate(r.t + dt)\n",
    "    index += 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot the results. The first column of the ```response``` vector is the position of the trolley, $x$, and the second column of the ```response``` vector is the angle of the pendulum, $\\theta$. We'll plot both below, after setting up plotting parameters to make the plot more readable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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yRwynYPfdKdhvX0K77abvS95TcZYOFWf+0fzGG2z60pchkWDIP+d+Zni5Hr24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\nu9IYM6ddMdblOSIiIiJ+5tuesxbGmCnAGpKPKz+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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11231ff28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the payload response\n",
    "\n",
    "# Set the plot size - 3x2 aspect ratio is best\n",
    "fig = plt.figure(figsize=(6, 4))\n",
    "ax = plt.gca()\n",
    "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n",
    "\n",
    "# Change the axis units to serif\n",
    "plt.setp(ax.get_ymajorticklabels(), family='serif', fontsize=18)\n",
    "plt.setp(ax.get_xmajorticklabels(), family='serif', fontsize=18)\n",
    "\n",
    "# Remove top and right axes border\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "\n",
    "# Only show axes ticks on the bottom and left axes\n",
    "ax.xaxis.set_ticks_position('bottom')\n",
    "ax.yaxis.set_ticks_position('left')\n",
    "\n",
    "# Turn on the plot grid and set appropriate linestyle and color\n",
    "ax.grid(True,linestyle=':', color='0.75')\n",
    "ax.set_axisbelow(True)\n",
    "\n",
    "# Define the X and Y axis labels\n",
    "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n",
    "plt.ylabel('Position (m)', family='serif', fontsize=22, weight='bold', labelpad=10)\n",
    "\n",
    "# Plot the data\n",
    "plt.plot(sim_time, response[:, 0], linewidth=2, linestyle='-', label = '$x$')\n",
    "\n",
    "# uncomment below and set limits if needed\n",
    "# plt.xlim(0, 5)\n",
    "# plt.ylim(-1, 1)\n",
    "\n",
    "# Create the legend, then fix the fontsize\n",
    "# leg = plt.legend(loc='upper right', ncol = 1, fancybox=True)\n",
    "# ltext  = leg.get_texts()\n",
    "# plt.setp(ltext, family='serif', fontsize=20)\n",
    "\n",
    "# Adjust the page layout filling the page using the new tight_layout command\n",
    "plt.tight_layout(pad=0.5)\n",
    "\n",
    "# Uncomment to save the figure as a high-res pdf in the current folder\n",
    "# It's saved at the original 6x4 size\n",
    "# plt.savefig('Trolley_Response.pdf')\n",
    "\n",
    "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot the suspension cable angle, $\\theta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Go3/254i/5RxM+d//UW3OmOj7xf/g2Cc/hYr3XIZJG+5UbU4QOWWB4Fgpq6lJ\nZu4C0EpEg/uU1A0VhAnBJ3PkCKxDh0HV1YjMmqXanDFBhoHYQneKKTytjoJG+nV7WjI2b14ggjAA\n9orbRAKZfftg9fSoNqds8VobBUAf5iJjjjrCHIgNNQUJALcAuNHdcDJnD5bEIqEojKSdyNWHkRGc\ny93TiYVAdB1U/ZHp3JCizg0qCFA0itj8+QCygeRYCKqPdKUYutRi+yg6txmIRpFpb4fV31/Ucwkn\nE5w70xghojlOZf2bASwnotuJaI37OTNvALCbiC4ioisAXMTMH1Nlr1Bcsk+mwRDqu8jqN/W4jddd\nfVhQ8FbdvibXjirSziKPIF07lEgg2twEMMOUfqUlJRj59nHAzG0Abhhlnw0lMkcoASMJJYNSUX8w\n2VpiwZ8mCGqNKjcbGaSbKZBfEB9UH+kIp9MwW1oAAFEnO+kHpfBRbOFCmK/vRHr7dsTPPrvo5xNs\nQpcRE4RcsrV8ghWIxRYuAACkd+4Ep9OKrSk/mDmb1QjQ1CQQriA+iJhtbUA6jcisWTCc1fZBQXRi\napBATAg8w2knuL/fbhNkGIgtWFBiqwrDqKmxFxekUjBbW1WbUxBB1B9l3jwA7umB0dAAY+rU0b+g\nEZ6+cMcOsGWN6TtB9JGueAH8Av+yYUBpfOTqId36eUJpkEBMCC3pnTuBTAbR5mZQZaVqc8aNVNhX\nh7kjK9QnyntVuhIikybBmDoV3NsrVdIVYDqLJKIBe/gDTg7ihdIhgZgQeIbTTrhi5aBNS7rEQjLF\nFET9UVD1YS7jDeKD6CNd8cqe+HztlMJHkRkzQDU1sI4cQebIkaKfT7CRQEwILX433S01XiAmq99K\nTnbFZDBaGw1Grh11uA9OQZNDAE6rI8fuoD8ABgkJxITAM5x2IrtiMpg306gj2A/6UvIg6o+8QGxh\n8G6mwPhF10H0kY5Y/f3I7NkDRKOINjf7euxS+cjTiYkkomRIICaEErasnKnJYGbEoo2NQDyOzBtv\nwDpxQrU5ZQOnUvYiD6JA6nwAIOoIxc1duxRbUl6Yu3YBzIg2NYHicdXm5IU3rS06sZIhgZgQeIbS\nTmT27QP39sKYOjWwvRopGrULLCLYWbGg6Y/M3bvt8gOzZ8OoqlJtTl5E580DiGC2toJTqVH3D5qP\ndCVb8sT/AL5UPpJArPRIICaEEq+ifkCnJV28djUBDsSCRlAr6udiVFYiMmsmYJp2XSuhJAR5xaSL\nqxEzX9/aDdlfAAAgAElEQVQJzmQUW1MeSCAmBJ6htBPZQq7BnJZ0cStzm68HNxALmv4oqIVcB5Pt\nOTn6tRM0H+lKMbWFpfKRUVcHY9o08MAAMvv2leSc5Y4EYkIoya6YDGbpChe3KKRkxEqHm9UI4qq3\nXLwgXq6dkpEt5hrsayc77ojGsBRIICYEnqG0E9mpyWAHYtH52WmCoBI0/VHaEbhHfa6MXmqy09qj\n30yD5iMdsbq6YB08CKqosLti+EwpfRSdNw+ABPGlQgIxIXRYXV3I7N8PqqhAtKlJtTkFEW2cDSQS\nyLz5Jqzjx1WbE3q4vx+ZPXvt8gONjarNKQhv5aTcTEuCm7WOLpgPikQUW1MY4wnihcKRQEwIPIO1\nE27ZiuiZCwM/IFIkgphTj8gM6KAYJP1RenerXX5gzpzAlh9wic6da6+cbGsbdeVkkHykK2aRpyVL\n6aPofCcjtkuC+FIggZgQOrKFXIM9LekSFZ1YyXBvPNF5cxVbUjhGZSUis2fZKycD3jg+CLhC/SCv\nmHSJuVOTu1rG3DheyB8JxITAM1g74a2YDEkgll399rpiS/IjSPojN+vo3oiCzlhXTgbJR7riPijF\niqQtLKWPjLo6GKdNtafqpXF80ZFATAgdnlA/4KUrXLxpAsmIFR1PqD8/HIGYrJwsHWbLbgDZf/Og\nE5vnXjvBlEQECQnEhMCTq53gVCr7ZBrggpy5xNyVkwEdEIOkP8pmxEJyM3UbOI9y7QTJRzpidXfD\n6uiwV0xOn16Uc5TaRyKJKB0SiAmhwmxx2tM0NsKYMEG1Ob4QmT0LqEggc+AArJ4e1eaEFk4mYba3\nA4aBaNMc1eb4gmRTS4O529bgRZubQUY4bqsxKWFRMsJxxQhlTa52IlvINRzTkoC7ctIWjwdxOXlQ\n9EdmWxuQySAyaxaoslK1Ob4Qa24GDMNeOZlMDrtfUHykK+mWFgBAdG5z0c5Rah+5QXxaGscXnbwC\nMSJaTkRXENHnnNflRLTMb+MEYbykvBWTwe4xORipCVV83CA3FhJ9GABQZaVdXDSTkZWTRcTc7ejD\nmosXiJWaaI5GTFZOFpcxB2JO8HU7EWUAbAFwD4Bbnde9AJ4nogwRfY+IGothrCAMRa52wnRqiIUp\nIwYEe+VkUPRHpifUD4c+zGUsbbKC4iNd8QKxImbESu2jSEM9jClTwH19yLz5ZknPXW6MKRAjotsB\nPA9gLQAC0AOgDcA259XmvEcA1gFoJaLvFcNgQRgOZvYyYvGQBWKSESs+YStd4RKGxvG6462YbA5+\n/blcsq2OZHqymIwYiBFRLRG1wA7AbgVwCYB6Zq5n5mZmXum8mpm5HkADgEsB3AZgHRHtIqKJxf4j\nhPLG1U5k3jwA7uqCUV8PY/o0xVb5S5BbjgRFf5R2i7mGaGoSGFtGLCg+0hE2TXuRB1DURR4qfORO\n06d3Bi8THyRGy4g9D2AT7ODrBmZ+iJm7h9uZmbuYeRMzXw87KHvYOYYgFJ3cQq5EpNgaf4nMnAkk\nErAOHoTV26vanNDB6TTM1jYATmugEOFpfZysjeAvmb377JXaM2bAqKpSbY6vSEasNAwbiBHRegC3\nMPN1IwVfw+EEZesA3EZEawoxUhBGwtVOZFdMhqOifi4UiXhP26azQisoBEF/ZO7ZY99MZ84M3820\nuSnbczKdHnKfIPhIV7wVk81NRT2PCh9ls6kSiBWTYQMxZr6NmTcUegJmvtOP4wjCaIStov5gYk6m\nRjIb/uM+8UdDpg8DnJ6TZ5xh95zcs1e1OaHDbHWF+uHKpAI5+sJdu8DMiq0JL+MuX0FEa4joflkZ\nKeiCq50IW4/JwbgDfTpgGbEg6I+8bgwh04e5uKv5zN1DXztB8JGuZIX6xS1docJHkUmTYDQ0gHt7\nZeVkEcmnjtgtsEX7I+ZhnXIX64noJiK6MC/rigQRzSGii1XbIfiH1duLTHs7EI8XdQm5SrybacAC\nsSBghqzH5GDcIMHcJdeO35QqEFOF151ht2Tii0U+gVgb7ECsmYjudl7X5u7g6Muehx20XQ9gExHd\nXbC1/rESwEYisoiok4geIKLlqo0S8qOjowPp7U79sPnzQfG4YouKQ3SuMyAGbGoyCPqjsPWYHIzX\nrmaYID4IPtIVN0CJFfkBUJWP3JIcQRt3gkQ0j+9cD+DBQe9dRUTXA1jJzD2wa4kBwK3MfAMRrQTw\nIBFdzsz/U4C9fsHM3EBEEx17hYDj6cNCVlE/F1cM7IquKRZTbFE44EwGabcg57zw6XyAbDY1LTdT\nX8l0dsLq7ARVV8OYFq6SOS4xycQXnXwyYhcD6IZdV+xq57UBQDOAG519JgFgADcBADO7xWDXDT6Y\nSiQICwdTpkzJ6sNCKtQHgiu61l1/lNm7F0gmYUybBqOmRrU5RSHqLfRoGVJ0rbuPdCXb2qip6CVz\nVPkoKouEik4+gdhaABc6dcXudV7rYBdyvdLZpxYABpW92AQ7WBME3wlz6Ypc3IzNcKJrYfykHd1U\n2Crq52JMmgSqqwMfPw7r8GHV5oSGbGujcGZSgZxsqow5RSOfQKyOmbcNfpOZN8FucTQkzNwFu8ir\nFhDRRTmv9URUq9omIT8OHziA9I4dAIDYmeGdmgSCKbrWXX9khnxaEgCIKFv+ZIhrR3cf6Uophfqq\nfBSZMQOoSMA6eAjW8eNKbAg7+QRirUOtgiSiiwCM1t+hLo/zFYOtAFqdTgEPwW5afo9im4R82bsX\nGEgicsYZMOp0ucSKw2iia2H85E4vhZmsTkyuHb/whPohXTEJOMWk5zj6VFk5WRTyCcR+AXsV5E1E\ndLnzuhmnCviRG7A5gVpr/qb6BzO3MXNb7jaAJlk5GUwmHDgAIPzTkkAwRde664/C2rB5MNlp7VOv\nHd19pCvetVOCkjkqfZQV7Adn3AkS4141yczXOzW4rh/00VbYJSEecLbbANxLRHfADsBuAbCxEGOL\nTBfsshanTLuuX78e1dXVAIBly5Zh9erV3o/CTRfLtrptc4vdzjS2aJEW9hRzu7u+HkBWdH3kyBGt\n7AvidqrFKV3R3KyFPcXadgPNvldf86YmdLIvaNucStmtsYgQbWxUbk8xtxPOtHb3iy/ixPnnKbdH\n1+18oXzbFhDRWtiBCwA8wMy/cN5/HrZYf53z+c05X2vOzUSpgIiaALQwszHo/QcA3D64vAYRsbR2\n0Js3r7oa/PQzaLjrTlRedplqc4oKM+PAkqXgri5M27oFkdNOU23SqHR0dGibccl0HsPBs5aCqqow\nfeeO0DWLz8Vsa8Ohd5yHyPTpmLbluZM+09lHupJuacHh8y9AZNYsTHvmqaKfT6WP+n75Sxz7+CdQ\ncdkfYdJd0rFwGPIePPKpIwbA7iE5zPsrczYfIqKtsKvwP6g6CHM4iqHLaKyCndUTAga/7rSnCXHp\nChdXdJ3asgXmrpZABGI6k1311hzqIAwAIrNmAfE4MgcOwOrthTFhgmqTAo2r0wxrJ49cpIRFcclH\nI3YKRDRxuM+YeZPT+FuHIGxwSQ0AXnbvbmZuL71FQiFkDh8Gjh4F1dQgMnOmanNKQtBE1zpnWtwy\nIGFtT5MLRSKINtnrqQbrxHT2ka54+rCm0izyUOkj928029vB6bQyO8JK3oGYU/ZhCxFZAI45760g\nohYiWuabhUWAmTc4JSvWOO2YJjLzx1TbJYyfbKPvM0Of0XAZSXQtjA9zt71+qBwCMUDa1fhJtrVR\nuBd5AIBRVWWXsUinYe7dp9qc0JFXIEZEG2GvklzhvgUAzLwVwHUAHiaiRh/sKxrMfBszb3D++1XV\n9gj54bU2KoNpSRfvZhqQWmKq6h+NBW96qUwCsdi8bIX9XHT2ka6kS7hiElDvI/fvlGLS/jPuQMzJ\nIF0Ju8XRJbBbHHk4hV03wF4lKQhFxauovyj8pStcpPebf7iZoXLIagDBm9bWFWbOqT9XHkG86MSK\nRz5i/WsAXOIUQgWAoaaEHoTepSqEkJB+bTuA8qgh5hI00bWu+iNOp2Hu3WuXH5jTqNqckjDczVRX\nH+mKdfQouLsbNHEijBL926n2kdfVQ4J438lnanJFbhA2DE3Qp4q+EFKs/n77qTQSQWz+fNXmlIyR\nRNfC2DH37AFME5EzzgBVVqo2pyR4N9O2NrBpKrYmuOROaZeLNjUmGbGikU8g9hARXTvKPldCSkEI\nRcbcsQOwLFBjI6iiQrU5JSVIomvV2pbhKJfWRrmcJLres9d7X1cf6Uope0y6qPZRbvNvqa3pL/kE\nYvcAuJOI7nfaGy0H7BIWzkrKBwBcDOBuPw0VhMG405K0oHyyYS7Dia6FsVMurY0GI6LrwsmumCwP\nfRgAGFOngmpqwF3dsI4eVW1OqBh3IOYUct0AW6h/L4AtzkfHALhB2FZmvs0vIwVhKFyh/oSVK0bZ\nM3wESXStWtsyHOV4MwWA6FyncXzOqltdfaQraQUZMdU+IqJsEB+AcSdI5FW+gpnXwV4t2YZsWX9y\nXrcy8yp/zBOE4fFKV5TRikkXWcFUOCqml3RAVt0Wjtla2tIVuhAkSUSQyLugKzPfy8zNTs/GuQAa\nmNlg5hv8M08QhoYtC+nt9tRkTxm2+QmS6Fq1tmUomBnp3eXToiYXN4hP59xMdfSRrvDAADJ79wGR\nCKKzZ5fsvDr4KBagTHyQ8KXFETO3MnNX7nsjtT0ShELJ7NkDPnECxrTTQA0Nqs0pObmi64xUuh43\nVmcnuKsbNGECjKlTVZtTUqI5+kIRXY8fs70dsCxEZs4EJRKqzSkpWX2hZMT8xJdAbDBEVAun7ZEg\nFIPcaUnV2glVuKv9dH861dE/uQ2by6X8gIsxeTKothbc0wPryBEAevpIV9y2WKUuAqyDj0QSURyG\nLehKRFcAyOdxaRJsIX95jW5CScn2mCw/fZhLdO5cJB9/wnk6vUS1OYHC6zHZVF7TkoAjum5qQnrb\nNpgtLYhocIMPErlBfLkRnT0biESQ2bcP3N9fNvX3is1IlfXvgR2I5RtQSc5bKBq5PSY7Ojq0eFos\nNW5GTPdpAh39495My23FpEtsbrMdiO1uReLtb9fSR7qiYsUkoMfviOJxRGfPhtnaCrOtHbFFZyq1\nJyyM1uKoG8DzQ7w/B3b1fHefVtiV9Oc477U6L0EoCuXYY3IwsoIpf7yMWJmtmHRx/27dp7V1pFxX\nTLpE5zbDbG1FuqVFAjGfGC0QW8nMbblvEFET7Nph1w9VK4yIrgRwJ+zq+oLgO5nOTmQOHABVVSE6\npxFTIhHVJikhFhDhrOqn+KFIl/H0EpArurYDUh19pCPMnC17UoYaMcD5ux94UPtxJ0iMJNbvBjBU\n+dzbAdw5XMFWZr4XwFoAtxRuniCcSvoVJxt25pmgMg3CAMCYNg1UXQ2rsxOZTlkbM1Y4mURm3z7A\nMBBtbFRtjhK88idSXX9cWIcOgXt7QXV1iJTham0gOJKIIDFsIMbM9czcM8RHqwA8N8pxtwI4pxDD\nBGE40q85gdiSxQD0qK+jAiIKxKCom3/MPXuATAaRmWeUXY9SF090vXcfeGBAOx/piqoVk4A+vyNP\nEqHxmBM08ilf0QbgxlH2WQugM49jC8KoeBmxxYsVW6IeyWyMn2yz7/LqMZkLJRKIzJoFMNt1sYQx\n4U1pl1Gj+MFkx5xWqUPnE6NpxIbibgA3E9EuAHfAFuW7xVybAawDsALArb5YKAiD8FZMOhkxXbQT\nKsgdFHVFN/9kWxuV780UAGLNzci0tcFs2Y0pCxeqNicQeEG8goyYLr+jSEM9jPp6WMeOwTp4EJHp\n01WbFHjGHYgx861EdA6AK5ANttyw2C11sUlaHQnFwOrvt0sPRCKILVig2hzleIGYrH4bM9lm3+Wb\nEQMcwf6mTTLFNA6ygVh5LvJwic6di9TmzUi37JZAzAfybfp9FYBLATwEW9RPAHpga8OuYuZLfbNQ\nEHIwd+wALAvReXM9fY8u2gkVxAJQ6Vo3/6QlIwYgt4TFbu18pCteNlVBIWCdfBQEbWqQyGdqEgDA\nzJsAbPLRFkEYFU8ftkj0YQAQndMIEMHcuxecToNiMdUmaQ0zK51e0gmvhEXrbmmDMgas/n5k9u8H\nolFEZ89SbY5SvFZHrfpKIoLEsBkxp1+kL/h5LKG88Qq5Ls4WctVFO6ECqqxE5IwzANO0VwNqiE7+\nsY4cAff0gGprYUyerNocpWSntXdjcpn/W4yFTGsbwIxoY6OSBx6dfkdeRkwkEb4w0tTkNUR0d6En\nIKKNAK4u9DiCAAApJyMWX7JEsSX6EA1IYVcd8PoENjWVXbPvwRgNDaC6OnBvL6xDh1Sboz2yYjKL\nOzWr8yKhIDFSHbE7ARhEtJmILhzvgYnoIiJqAdDJzBsKMVIQAIAzGZjbtwM4OSOmk3ZCBd6gqKlO\nTCf/ZOtAlbfYGrDr0LkawyNbtyq2Rn/caThVU9o6/Y6is2cB0Sgyb7wBq79ftTmBZ0SxviPK3wpg\nkxOQ3URElxNRIxFNdPcjoolENIeIriCim50A7EHYqyevK+6fIJQLZmsreGAAkRkzYNTXqzZHG4LS\n6kgHslkNCcSAbHaH29rVGhIATLl2PCgWs4sCw5myFQpiVLE+M68jogcBbACw0n0bwFCpffeNLtir\nJ3/hk52CMKQ+DNBLO6GCXK2Pjujkn3Jv9j0YN7tTceiwYkv0J1t/Ts21o9PvCHCaf+/ejfTu3aeM\nycL4GFP5Cma+l5nrYWu9HoIdcA1+9cBeRXkVMzdIECb4jbdiUvRhJ+FqxNK7W6TS9SiYrVK6Ipds\nGQIRXY8EW1a2/pxcOwByi0nr+QAYJMZVR8wJyC5hZgNAA4C5zqvB6U15qQRgQrEYLiOmk3ZCBcbU\nqaAJE8Bd3bA69esspot/OJlEZm95N/sejNvmKblzl2JL9CZz4CC4vx/GpEnKZBG6/I5cpJaYf+RV\n0BUAmLmLmVudV9fo3xCE/GFmyYgNAxFlV07KcvJhMffsASzLbvadSKg2Rwtc0TUOHACL6HpY3Ixh\nuVfUz8Vr/q2pJCJI5B2IBR0iWussLriCiNartkcYGevgQVidnaC6WkRmzDjpM920EyrwBkUNl5Pr\n4h9v1Zvowzw80TUzTBHsD4unLVRYBFiX35FL7tSkSCIKoywDMSJaC8Bi5l84U6mbiOh21XYJw5PK\nqahf7vWfhiImBRZHxbuZNonGJxdPYyjXzrDk1p8TbCIN9TAaGsB9fbAOHFRtTqApy0AMwFpmvsvd\nYOZtAC5WaI8wCsPpwwD9tBMq8FqOaJgR08U/WbG1ZMRyEdH16LjTbyobxevyO8rF61cq105BlF0g\nRkR1AFYM8VEXEV1UanuEseEGYlJRf2hc4axkNYZHSlcMjXRmGJ1sf1K5dnKRa8cfyi4QA9AEu87Z\nYDoxdIAmaMBIGTHdtBMqiDY2AoaBzL594GRStTknoYt/vJtp0xzFluiF7p0ZVGOdOIHMgQNAPI7I\nzJnK7NDld5SLZFP9wZdALLfKfgBogB10DaYLwKQS2yKMAaunB5k9e4F4XKlYVmeoogKRmWcAmYy2\nzb9Vkuk8BuvYMVB1NYxp01SboxUiuh4Zb5HHnEZQJKLWGM2QEhb+MGpl/eFwpvFugZ1FYgARIloB\nYCOAK5n5BX9MLClDjkJvNs8DDCPnRSA6eRs06L2IgeicOYivXInK91yG2IIFpf5bQkP65VcAALFF\nZ4JisVM+7+jo0PJpsdREm+cis2cvzN27EZs/X7U5Hjr4J5sNk2bfg4k01AP1deBjXbAOHETk9Omq\nTdKKbGsjtQ+BOvyOBiMlLPwhr0CMiDYCuDL3LQBg5q1EdB2Ah4loBTO3F25iUWgY4r06AEeH2pkH\nBk59bwwnybTvQfKRR3H8q19DbPly1Hzi4zi+fDmIyPtBuQJM2R5+O/P00wCA+FlnDfl5T0+PVvaq\n2o42NyH58MPofvElVF52mXJ73G0d/FPlVNTPzJhx0g1Nh38fHbapsRF87AUc2fo8jNhblduj07b5\n0ssA7OyPDvbotH2sssJu/r1/P6z+fhzt7dXKvlJv5wuNNxXt1Ny6BcCtsBt71wPY6FTbd/e5BUAj\nM19TkHVFwBHrd+ba67y/BcDnmfnhQe9zpq8PsKyTXsw86D0Gs/P/zOBkEuaO1zHw+OPo/+3vwN3d\nAID429+Oupu+gti8eSX7m4NO58f+Fv2/vg91X70N1R/6U9XmaMuJH/8EXdffgMorr0TDN76u2hyt\n6P63m9D7ne+i5rOfwcTP/L1qc7Tj2PrPo++nP0PtV/4VEz76UdXmaEXndR9D/32/Qf03/gNVV16h\n2hztOHT+BTBbWjD1gfvLvedk3qn2fDJi1wC4hJkf8s5+aqr/QdhTlNrBzF1E1EpEtczcnfNR3eAg\nzMWorMzrXLF581D5vvei9ktfRN9Pf4bj3/gmUs88g8N/dBnqvvJlVP+pBBVjIeU8kcbOOkuxJXoj\nK5iGx5ualD6BQ+JpfWSK6RTSLXLtjES0uQlmSwvSLS3lHojlTT5i/RW5QdgwNMGe6tOVWwDc6G44\n2rYHi3Uyo6oKE679G5z2+KOouupKYCCJrs+ux7HPfBacThfrtKHA6u5Gpr0dSCQQWzC07knH+joq\n0FV0rYN/pHTFyPQ5UysSxJ8MWxbMNj2uHR1+R0PhjTut+tUwDAr5BGIPEdG1o+xzJYCteRy7JDDz\nBgC7iegiIroCwEXM/LFin9eor0f9f3wddf/+NVBFBfru3oijf/03sKTH27CMJtQXshiTJ4Nqa8E9\nPbCOHFFtjjawacJsbwcgldGHg2Y3ApCM2GAy+/cDA0kYp02FMTFIxQFKh/S5LZx8ArF7ANxJRPcT\n0eVEtBywS1g4gc0DsKvU3+2noX7DzBuY+SGnzdFtpTx39TVXY/L/3AujoQHJhx/B0Q99GNaJE6U0\nITCkXranJeMjTEvqtpJIFUTkBRo6DYqq/ZPZtw9IpxGZPh1GVZVSW3RlyrKzgVgMmTffhNXXp9oc\nbci2NlKfSVX9OxoOrw6dhl09gsK4AzFmvhPABgCXALgXwBbno2MA3CBsa6mDm6ARP/tsTP7fXyBy\n+ulIbd6MzmvXaFeIUwfSL74IAIidvVSxJcEgNlcGxcHItOToUDRqFwWGTDHlkm1tJNfOcOgqiQgS\neRV0ZeZ1AK4G0IbsSgFyXrcy8yp/zAs3sblzMXnjz2FMnozk40/g2Cc/Dc5kVJulFdmM2PCBmK7a\nCRV4vd80yoip9k9ahPqj0tHRIYs9hiDb2kh9IWnVv6PhkObfhZN3ZX1mvpeZm50yEHMBNDCzwcw3\n+Gde+InOmYNJP/kxqKYG/b/5DXpulUSii9XVhUz7HiCRQHS+lPsYC1HJiJ2CZMTGhpfZEJ2Yh6yY\nHBvS/LswfGlxxMytzHxS/0YiWubHscuB+JLFmHTXBiASQe+3v4O+X/1atUlakPKE+otGFOrrqp1Q\ngfvkbu7WJyOm2j9mq9xMR8MuCCyi68HolBFT/TsaiewDoFw7+VDMpt+jlbgQcki8YzVqv/jPAICu\nz3wW6de2K7ZIPWl3WnKp1A8bK9HZs4FIBJm9+4bsCFGOSEZsbMS8IF6yqYDd49Y6fBioSCAyY4Zq\nc7QmqxOTaycfhi3o6lTQz1d5Nxl2xX1hHFT/1UeRfuUV9N29EZ1/+3FM+f1v8y4mGwbSL70EYHSh\nfm7LmnKH4nFEZs1Cpq0NZns7YgsXqjZJqX+s48flZjoGOjo6MCmngTNbFsgo5nO6/njZsDlNWvxb\n6DzOSTa1MEaqrH9LgceW5RPjhIhQ+5UvI7V1G8xdu9D9xX9B/S03qTZLGSknEBtJqC+cSqy52Q7E\nWnZrEYipJHsznaPFzVRnjNpaGFOmwOroQObAAUTLPHCVFZNjRzJihTFai6OtAJ7P47gNAC7P43tl\nj1FZiYbvfBuH3/s+9P34x6g4/zxUvucy1WaVHKurC5k9e4GK0YX6uj4lqiI6txnYtEmb1W8q/eNN\nS2pQB0pnXB9Fm5uQ6uiA2dJS9oFYdrWtHteOzuNcdNZMuw6d0/y7nGdy8mG0QOwqZm7L58BEZOXz\nPQGILV6E2n/4Arr/6Z9xbP3nET9nFSIa/wiLgZsNiy1aDIrm0xK1fMmWsNAjEFOJG4zGRKg/JqLN\nc5H6v2ftAPb881Wbo5SsUF+PQExnKBZDdPZsmC0tMHe3Ir5ksWqTAsVIufoNADoLOPbVBXy37Kn+\n679C4vzzwF1d6P7Hf1JtTslJbd0GAIgvXz7qvrrW11GFt4KpVY9ATKV/3OKkumQ1dMX1UUza1Xjo\ntGIS0H+ci+ZoDIXxMWwgxszrmLl78PtEtMZpb9Q40oGZ+d7CzStfiAh1t9wMqqpC/32/Qf8f/qDa\npJKS3vYCACC+QqqgjJfcelDlXuk6u2JSMmJjQWqJ2bBpwmxrByD9ScdKtnROeV87+ZCPevUW2O2N\nRrw6iWg5Ea0nopuI6MK8rCtzojNnYuKNdn3crv/3BVjdp8TFoYSZkdo29oyYztoJFRgNDaC6OnBv\nr71iUDGq/MOWJRmxMeJpxOZKYU7A6U+aStn9SaurVZsDQP9xTjJi+ZNPINYGOxBrJqK7nde1uTs4\npS+ehx20XQ9gExFp3QRcV6r/8iOIr1oF69BhdP/bzarNKQmZfftgHT0Ko6EBkVmzVJsTOIgoWxOq\njDMbmQMHwAMDMKZMgTFxompzAkHkjDOARALWwYOwentVm6MML5OqybRkEIg2Sx26fMknELsewIMA\n7gBwlfO6k4h2EZE72q1z/nur0wLpHACXEJGspBwnFImg7rZbgGgUfT/5iSdiDzNuNiy2fDmIaJS9\n9ddOqMB9OtWh56Qq/3gan6Y5Ss4fJFwfUSSC6JxGAOXd/Nv93eg0pa37OBfLrUNX5pKI8ZJPIHYx\ngG4At8IW5F8NW9jfDOBGZ59JsOuI3QQAzPw8gLXIBmjCOIjNn48Jf/PXADO6vvCPYCvcC1KzQn3R\nh1oGAucAACAASURBVOVLVKqkS0X9PHFLfZRzNlU3oX4QMOql+Xe+5BOIrQVwITPf4DT+vpeZ1wG4\nFMCVzj61ADBI7L8JdrAm5EHN338axtSpSG/dir57wr0OIivUH10fBuivnVBBVq+hPiOmyj/ezVSj\nrIau5PrIWzlZxlofU7MaYkAwxjlPY6hBJj5I5BOI1THztsFvMvMmAMPOIzlNwRvyOJ8AwKipQe0/\nfAEA0PNvN4VWuM+pFFKv2M2+48skI5YvotfInZrU52YaBKKiL/T+dp0CsSDgrbrVpHROUMgnEGsd\nahUkEV0EYDQxRl0e5xMcKi//E8Tfcg6sI0fQ87WvqzanKKS3bweSSUSbm2HU1o7pO7prJ1QQnT0L\niEaReeMNcH+/UlvUacRkanKs5PrI0xdqkE1VgXXsGKyjR0FVVYhMn6baHI8gjHNeDcMyDuLzIZ9A\n7BewV0HeRESXO6+bYQv4TyI3YHMCtfJ9PPcBIkLdl78MEOHED34Asy2vpgdakyvUF/LHrXQNZq8e\nUjlh9fcjs38/EI3a7VeEMeNlNdrawJmMYmtKT9pri6VHs+8g4ekLy3haOx/GfZUx8/UAtsFePXmv\n8/o87L6UNxLRA86ubQDudQK2NQDuga0TEwogtngRqq6+CjBNdN9UaF92/UhtdfRh4xDqB0E7oQJd\n9Boq/JNptR9SorNng2Kxkp8/aOT6yKipgXHaVGAgaQezZYarq9SttVEQxjlp/p0feYX7zLwSwHWw\nV0tugN2TchUz3wp7xWQr7BWSN8MO2O6APS0ZvshBARM/9zlQRQUGfvtbJLfk05NdX1JbtwIYu1Bf\nGJ7soFh+T6dpEeoXRFZjWH7XjqcPkxWT4+ak5t99farNCQx5512Z+U6nDdI6Zv5FzvsrmXkuMz/k\nBGaXwg7amvNtIC6cTOT06aheY9fQ7fnXL4emZkvmyBFk2tpAVVWILVo05u8FQTuhgqgmq99U+EfH\nVW86M9hHXk2oMtT6ZBd56BXEB2Gc8yQRAMxWud2PlaJNgBPRLsBeTekEbeIVH6n5+N/CmDQJqS1b\nMBCSPpSpzZsBAPEVK0DRqGJrgk8514OS1kaFUc59A2XFZGFkHwDLc7FHPhQUiBHRHCJaNsTrCkjN\nsKJi1NSg5jOfBgB0f+UmcDqt2KLCST3nBGLnrBrX94KgnVBB7tSkyqypCv9IDbHxMdhHWX1heQVi\nnE7D3LMHIPKygroQlHFOdGLjJ69AjIhuJ6IMgBbYPSUHv+6BXVlfKCLVH/4wok1NyLS14cRPf6ba\nnIJJbt4CAIi/5RzFloSDSEN5VrpmZildUSDlWg/K3LMHME1EZs4EVVaqNieQlLM2NV/GHYg5pSrW\nwi7e2gOgfYhXOKuNagbFYph4/ecBAMe/+U3l9aIKwervR/rllwHDQHzFinF9NwjaCVV4mQ2Fg2Kp\n/WMdPgzu7QXV1cJokBrSY2GwjyIzZgAVCViHDsPq6VFkVekxd+0CoN+KSSA445wXiJVZNrUQ8smI\nXQWgC8BKZq5n5uYhXvUYocq+4B8V77kMsSVLYB08hN4f/FC1OXmT3vYCYJqILVoEY8IE1eaEhuzT\nafnoNbxsWFPzmJrGC6dChoHonGwT53JB9GGFc1Lz75D3RfaLfAKxegA3DdXmaBA35HFsYZyQYWDi\n59cDAHq/811Yvb2KLcoPT6ifx7RkULQTKsgKZ9XpNUrtH9Opm6abxkdnhvJRTINrp9SkdznXzrx5\nii05laCMc0Z9PYxJk8D9/ciUkSSiEPIJxJ4HMJYR7o48ji3kQeLCCxBftQpWZyd67/pP1ebkRdIN\nxFaNT6gvjIxXD6qMmvC6AvOohjfTIOFmhVQXBC4luhZzDRrR5vLLphZCPoHY9QCuGarf5CDK5zFK\nMUSUzYrdcSesri7FFo0PzmSQcgrTJs4Zf0YsKNoJFcQ0WMFUav+YLY7OZ54U5BwrQ/lIh2xqKWHm\n7NSkhkF8kMa5ci5/kg/5FGtaCWAL7H6TD8IOuAaXd58LewpTO4hoDuzisqFqt5RYfS4S73gHkk8+\niePfux21NwZnZth8fSf4+HFEZs5E5PTpqs0JFZFBla6NqirVJhUd051emqvfzTRIZG+m5ZERsw4e\nBPf2wqivR0QWeRSEZMTGRz4ZsTsAXOz8/yWwWxndOej1eehbvmIlgI1EZBFRJxE9QESh6KfjrqA8\n8Z//hUyAnp6Szz0LYPz1w1yCop1QAUWjiDY2AsgWOS01pfSPdeIEMm++CcTjdhAqjImhfORWljfb\n2sui+Xda89ZGQRrnyq3594kf/qig7+db0LUVwEMjvHSuos/M3ACgjpkbmPnSMSw8CATxFctRccnF\n4P5+HP/Wd1SbM2aSTz0DAEi87W2KLQknurQ6KgVeIdc5jdKdoUCM6mpEpk8HUilk9u1TbU7RcXWU\nMqVdODE3m1omJSxSL71U0PfzDcQuYeaRXs3QvHwFM4eyOM7E9bZW7MSPfgRz/5uKrRkdtiyknnEC\nsdXn5nWMIGknVKC6rk8p/eNOS7qLFISxMZyPVF87pcQLxDQtXRGkcc6TRLz5Zlk0/3bHnXzJJxC7\nbox9I6/K49hCgcQWL0Ll+98HpFI4/s1vqTZnVMztO2AdO4bI6acj4jSLFfylnCpde6UrJKvhCzoU\nBC4VbrCpY+mKoKGDJKJUMDPSzgKhfBl3IMbMd462DxE1AdiYl0UlgIguynmtJ6Ja1Tb5Sc1nPwMY\nBvp+/nO7ZYfGJJ9+GgAQP/fcvItvBkk7oQLV0wSl9E9appfyYjgflVMQ795MdS1dEbRxrlwE+9bR\no+CuwpoJFUtE4bZA0pGtAOBm9YioFXZvzEuH+8L69etRXV0NAFi2bBlWr17t/SjcdLFW27W1qLri\ncvTdcy86broF0X/9kl725Wz3PPIIgOy0pGp7wrjNtfZzhtnaisOHDoEMQyv7/Nwe2LEDQFZwrdqe\noG+fmDwZQPZmqtqeYm1PqqiAdfAQkEigM5HAVEAr+4K47QbxPS+9jKoPfEC5PcXatrYMLhoxfojZ\nn8WNTlZpLexVlE2wRfERXw5eZIioBcBVQ4n2iYj9+jcqJeaePTh03rsAy8LURx7ysiI6waaJA0uW\ngo8fx2nP/R+iM2bkdZyOjo7APS2WmgPLVsDq6Cjo3zlfSuUfTqfx5rwFQDqN6bteL4tSHX4xnI/M\n/W/i0FveCmPyZEx/MRRrmoYktW0bOt77fkTPPBOnbXpAtTlDErRx7sTdG9H1mc+i8gPvR8N3v6Pa\nnKJx4oc/QteN/w8z9u/LO/lUcEaMiC6CXeT1IpQwC0ZEazB2HdpVzDxS7rALdlmL0Iw00dmzUXXN\nNej7yU9w/Gv/jobvfVe1SaeQfuUVu37Y7FklDw7KjWhzE1IdHTB37w7tv7W5Zy+QTiMyY4YEYT4R\nmT4NVFkJ68gRWF1dMOrqVJtUFDx9mKbTkkEkqkEx6VKQ9kHykdeqSSKaQ0Q3E1EngAdh1xVzg7BN\nKEFAw8wbnNITY3l1O3Y3EdFQXUg7nVeoqPnUJ4F4HP2/vg/p17arNucUkk/Z+rDE6tUFHSdIT4mq\nyLY6Kr1eo1T+8drTiD5s3AznIzKMsrihZrWF+gr1gzbOlUvzb7NAoT4wzkCMiNYQ0RYALbCLtrqP\nR1thZ8XqmflS2JkqHTViR2FPnQ5mFRztWJiIzjgd1X/x5wCAnq99TbE1pzLw6GMAgMQ7CgvEhNHx\nhLMh7hvola7QcBo+yLjXTph7TnqlKyQj5hvl0vy70NIVwBgCMSJaTkS3E1EGdlX9FbCDrC4Atzm7\nrWHm23Km/zqh4TTfUNOTRLQWwN3M3F56i4pPzd99HFRRgYE/3I/Uiy+qNsfD6u1FavNmwDCQeOd5\nBR0rSPV1VOGtnFSQ1SiVf9JeayMJxMbLSD7yWh2FuAxBdmpS34xYEMe5bDHpcAbxXiePWKyg4wwb\niDllHVpg95HMXQV5L+yCrg3MfL3z3klqdmbuYuaVBVlWJJh5g/O3rSGi9QAmMvPHVNtVLCJTp6L6\nrz4KAOi57atqjckh+fTTQDqN2LJliDRo2ZY0VJRFVkOmJotCtqhrOK8dTqdhtrcDRIg2zVFtTqgI\n+7S2l0kt8LoZSax/HQD36F2wpx43jiJ6DwTMfNvoe4WHCX/7MZz44Y+QfORRJJ97Dom3vEW1SUg+\n8igAoOLCCwo+VtC0EyqIzJwJxOOwDh6E1dsLY8KEkp27JCsmmWVqsgBG8lHob6Z79gCmicisWaDK\nStXmDEsQx7mw1xIzvf6khWVSh82IOW2KVgHYAFsLtg7AGiKaWNAZhZITaWjAhDXXAgB6blWfFWNm\nDLiB2LvOV2tMmUCRiPfUFsYpJuvgQXBvL6iuDsakSarNCRXezbS9HWyaiq3xH9GHFQ+Vi4RKQXqX\nLdQvtJPHiBoxZt7KzOuY2QBwM4B3A+giovuJ6PKRvktE3yvIMsFXJqxdA6qtReqZZ5B88imltpi7\nW5HZtw9GfT1iS5cWfLwgaidUEG1SUyW9FP5J57SnybdDQzkzko+MykpEZswA0mlk9oav+Xe2P6ne\ngVgQx7lYyDsz+NUofsyrJpn5Xma+BEADgIcA3OqUrwDsAq4eTm2xoVYnCoowamtRc53tkp5bb4PK\nIrXJRx8FACTOPw8UCUTN31Dg1kgK49OpZDWKi9dzMoQ6sbT0mCwaYW/+nZVDFGlqcjgcIf6tzDwX\ndv2wDQDuIqJOIvoeEX0OwC0YJOAX1FP9N38No6EBqeefR/LhR5TZMfDgJgBAxQWF68OAYGonVKCq\nb2Ap/GPucvsEij4sH0bzUZh7Tpqa95h0CeI4F+bm3yct8mhuGnX/kciroKtnSHbqsgH2ysq5AG6F\nXeJC0AyjuhoTPv63ANRlxTKdx5B85hkgEkHFRReW/PzlTDarEcabqVt+QAKxYhDWQIyZs4JryYgV\nhbDWMPQWeZxxBowCF3kUFIjlkjN1uQoa1hATbKr/8iMwTpuK9CuvYOD3fyj5+Qc2bQIyGSTOPRdG\nvT9lK4KonVCBdzNtay1ppevSaMScrIaUrsiL0Xzk1aELWRBvHbAXeRj19Yg0NKg2Z0SCOs55deh2\nFl6BXifM13cC8CeA9y0Qc2HmrQDWQM/K+mWPUVmJmk9+AgDQ89WvgjOZkp5/4Pe/BwBU/NG7S3pe\nATBqamCcNhUYSCKzf79qc3zD6u6GdegwUJFA5IwzVJsTSsJahiC983UAQHTBfMWWhJfYggUAgPTO\nnYot8Rf374n5cO34HogBXjAmj6aaUv2hDyEyYwbM13ei/777SnZeq68PA48/DgCo9DEQC6J2QhXZ\n5eSlmyYotn+8AXHefFn8kSej+ciYNg1UXQ2rsxOZzmMlsqr4pJ2sRmy+/oFYUMe5qPNv62aQwoL5\nuhPE+3DtFCUQAwBmDpcyL0RQIoGaT38KANDz1X8Hp9MlOW/ykUeBgSRiK1YgMm1aSc4pnEy2EW94\nfp7eFEEAbqZBhXIEyWHKiplOEC8ZseIRm9sMGIZdh25gQLU5vqF9RkzQn6qrrkSksRGZtjac+MlP\nS3LO/t/9DgBQ+Z7LfD1uULUTKvD0GiXMiBXbP2nnydSPAbFcGYuPvGsnRH0DsxmxBYotGZ2gjnNU\nUYHI7NmA9f/bu/Mwqao7b+DfX+29QDcNopCI0iyyiLK5L1FBJTOZbG7JxGQmmVfQxMz7zBgEZ5JM\nZhJHULNNJgmKySTO88YoYpKZuARQYwwoYXVhU1YNigJNd9Nbbfe8f9x7q4umuruWu9f38zz9QHXd\nOvfAqbr3V+f8zjlaYL4AqnQamb37AHg0R4z8QaJRNPzzPwEAjn/7O9Da2209n9bZiZ7frQJgfSBG\nxTOT2c0VoYOAPWLOiDSbs9+C0SOmb4tlTPJgEG8r80uSmZPnd5l9+4B0GuExYxCqra24PAZiVSzx\n4XmInX8etJYWHP/hj2w9V8/Tz0B1dyN23nmInHGGpWX7NXfCDbnE2Z27HFu+xLEcMd5My1ZMG+Xe\nOwHJ9cm+844+Y3L4cIR9sC2Wn69zZg5eUGZOZizOLWQgVsVEBA1f+xoAoGP5Q8jYOJOu64knAAC1\n1w24MxbZLHTqqZCGBqjWVmjvv+92dSqWbTkG7fBhSE0NZ0zaLGIEYmaSst+xJ9U5kVyPWDCC+LTF\nuYUMxKpcbOYM1Hzso0AyifYl99pyjux77yH54h+BaBQ1H/lLy8v3a+6EG0Skd5jAoZ4NO9snk7f8\ngIR4OStXUTliZ54BxOPIHjwI7fhxB2plL3OYzC89qX6+zpk5eEHpTbV6ti2vXIShixcBsRi6n3gC\nqVdftbz8rl8+CmgaEnPnWLaIK5UvGqCeDT8tP+B3Eg7n9mMMwg2VPWLOiYxrBsJhZA8cCMTMSatn\n2zIQI0TGjEH9Fz4PAGj7+jcszR1S2WxuVmbdzZ+xrNx8fs6dcENkkvnt1JlAzM724fID1ii2jYI0\nPOm33EI/X+ckHtf3nNQ032+xplIpPVlfxLK9bRmIEQBgyN9/GaERI5DasAHdj6+0rNyeZ59D9uBB\nhM88A/HLL7esXCpfbphgZwBupj5afiAIog4H8XZRmpZLHI/wveMI88tSxud5Ypm9e/U9JsecXvEe\nkyYGYgQACDU0oOGr/wwAaPvW3dDa2iwpt/PhhwEAdZ+92bYcHj/nTrgh/4LoxMxJW3PEdnGLGisU\n20a5YW2fB/HZgwehuroQOuUUhJv8kS7h9+ucmT7g9yDejnQIBmKUU3P9dfpyFkeOoP2++ysuL71t\nO5LP/x6SSKD2xhstqCFZITx8OEKnnALV2enrPSezR45Aa2mB1NcjPHq029WpCk4Pa9vF6uUHaHC5\nrY783iOWS4ewrieVgRjliAga774bCIfR+fOHkXr99YrKO/6DHwAAaj/zGYSbmqyoYkF+zp1wi5M9\nG3a1T36ytYjYco5qUWwbhUePhtTXQztyBNkjR2yulX3SPuxJ9ft1LhqQJSxyuYXsESO7RKdMRt3n\n/xbQNLQuvgsqmy2rnPSevej+7ZNANIohty6wtpJUsdy6Pj7u2fBbsnUQiEggNnHmbFvnRZqbgUgE\n2QNvQevudrs6Zct9AbTwusNAjE4y9Ct3IHTaaUhv2YqO5cvLKqP9nnsApVB74w0Ijx5lcQ1P5Pfc\nCTfkr7BvN7vax8wP4820cqW0URAS9v0429bv1zmJxRAZOxZQytG9bq2kkklk9u8HQiFEx42zrFwG\nYnSS0JAhGHbvUgBA+333lzzdOPnHteh5+hlIbS2G/uM/2FFFqlBuGQIfDxNYvbo1FcfJIN4OStNy\ne0wyiHeW33tTM3v2AtkswmPGQCyaMQkwEKN+JOZchdobrgd6kmi94ytFD1GqVAqt//IvAIAhX74d\n4dNOs7OaAPyfO+GGXL7Gm2+WPfxcLDvaRynF4SULldJGfl9LLPvWW1A9PQidOhKhxka3q1O0IFzn\noj5PibBrNwYGYtSvhm/8C0KnjkRq40Z0/HhZUa9pv//byOzchfAZY1A//xaba0jlCg0Zos80TCaR\n2X/A7eqUTDt8GKq1FTJ0KEIOBPvUKzc06dDyJ1ZL79wJoLdnj5wTnTwZQG8b+E16h/HeMf4dVmEg\nRv0KNTZi2P36Mhbt996Hnhf/OODx3U89jY4f/ggIhTDs+9+DJBJOVNP3uRNuMZcisLtnw472SW/f\nDgCITp7EGZMWKKWNQiNGINTUBNXeDu3dQzbWyh65m+mUKS7XpDRBuM5FJ08CAKS373C5JuUx681A\njByVuOpK1N/+JSCbRcvf/R8k168veFzP88/j2Jf/HgAwdNGdiJ93npPVpDLkcn18OExg1zdTGpyI\n5IYn07v817Nh182UBhc+4wxITQ20Q4eQbTnmdnVKltlhvHcsDuIZiNGghi66EzWf/ARUZyeO3PRp\nHP+PH0Dr6AAAaK2taF96L47+zeehenpQe/PNqP/SFx2tXxByJ9zg1ObfdrRPrkfMZ70aXlVqG/l5\n5mR6hz8DsSBc5yQUQmSS3iuW8dnwpHbsGLLvvgupqUH4jDGWlh2xtDQPEZGZABYrpU5a0l1E5gM4\najxsVkrd52jlfEZCIQz77ncQGtaEzp/8BO1L70X7d7+H8CmnIHvoEJDNAiKo//LtGLroTg4V+URu\nlfQd/rogAv69mQZFrjd1h78CMa2rC9n9+4FIBJEJ1mzYTKWJTpmM9JYtSO/YgfjFF7ldnaKZeW2R\nSWdBwmFLyw5cj5iIzBCRJQBuBNBc4Pn5ADSl1Eql1EoAa0SkuEz0KiaRCBr/7RsY/sgvELvwAiCd\n1rfHUQrxyy7DiJUr0LB4kStBWBByJ9wQnTABCIeR2bsXysYFFq1uH5VKIfPmbkAkF0xSZUpto1yv\nxg5/5fpkdu4ClEJkwnhILOZ2dUoSlOtcLmHf6NX2CzuHtAPXI6aU2gJgi4jMADC3wCHzlVKz848X\nkULHUQGJyy9D4vLLoB0/Du3oUYROPdWyHejJWZJIIDJ+HDK73kB61y7Epk93u0pFyby5G8hkEB47\nFqHaWrerU5VySddvvgmVSvkmqGFPqvuiU4xAzGdBvJ3vncD1iOU5qWtGRBoBzCxwbKuIzLG/SsER\nGjIEkTPP9EQQFoTcCbeYOVbpbfZ9O7W6fXgztV6pbRSqr0f4zDOBVAqZEhd8dlPapmRrJwTlOhc1\nelPTu3bZvoahlXqvO5MsLzvIgVghzQBaC/y+BYUDNKJAywViPhom6E3UZyDmJl+/d2y4mVJxQg0N\nCH/gA0BPEpl9+92uTlFUNqsPa4M9YlZogh509dUKYLjDdSGLBCV3wg3RqfbfTK1un95eDQZiVimn\njXJDTNu2WV0dWyilfL3sSZCuc+b/v19yDDP7D0D19CA8erQtuzEELkesAv0uEb1w4ULU1dUBAKZP\nn45LLrkk101sfjj42L3H7e3tnqqPnx63n3oqAD0RVWkajhw9avn5rG6f1Ov6jT86ebLr/39BeWwq\n5fXRqVMBAJ2vvIKGMl7v9OPsO+9AtbcDwxoRGjnS9fpU8+PY5EnAmjVo3bQJHRde4Hp9BntcbwSM\n2rhmHD58uN/jyyVe3aJCRG4BcEORh9+glGrr8/qZAB7MT8w3kvIfU0o19Tl2FYBVSqn7C9RDefX/\niMgK706fCe3wYZy67o+InHGG29UZUPbwYRyaPhNSX49RO7dzqRQXZQ4exHvnX4jQsGE47bVXPN8W\n3avXoOVvP4/4pZdixKOPuF2dqtb1m//BsS9+CYlrrsbw//qp29UZVPt99+P4976P+tu/hIa7Fvd3\nWNkfAM/2iCmllgNYbnGxGwEU6ldsArDZ4nMR+UJ06hQkf/8C0tu3ez4Q683xmez5G3/QhUePhjQ2\nQDt2DNqhQwiPGuV2lQaUMd47EeaHuS43rO2TrY7sToeoqhwxpVQrgL0i0tDnqUal1HNu1IkqF6Tc\nCTfYPXPSyvZhfpg9ymkjEUF0sv2zbq3i99m2QbrORcaOBeJxZP/8Z2htbYO/wGV2b4sV5ECsqZ/f\nLwVwl/nAGMJc7UiNiDzIiYR9q6S3+ftmGjS+eu9sZxDvFRKJIHrWRADeX09Ma21F9u23gUQckeaT\n1oi3ROACMREZa6ysvwTADBFZZuSbAcgNee4RkTkich2AOUqp29yqL1UuKOvruMXuHjEr2ye97XUA\n/lwHysvKbSMn1qGzgtbRgczevUA0iujEiW5XpyxBu85Fp00DAKRffc3lmgwsnZscNAUSsSeby7M5\nYuVSSu0D0G82nXGM1blnRL4VaW4+YZgg1NB35N4btO5ufVX9cJi9Gh5hzpz0eo9Yets2QClEJ02C\nxONuV4cAxKZNQxeA1Guvu12VAaVe1wPF2LSzbTtH4HrEqPoEKXfCDRKJIJrbANz6YQKr2ie9bTug\nafo+gR7Y0SFIym2j6ITxQCSCzL590Lq6LK6VddLGzT5q483UbkG7zpltkX7d24FY7r1zzjm2nYOB\nGBH1DjEZ3fBelM59M53mck3IJPE4IhPGA0p5egac2esSPdu/gVjQRCdN0oP43burPohnIEa+F7Tc\nCTeY+RopG/I1rGofM5fEzm+m1aqSNopO9X7PRvq1VwEAsXP8G8QH7ToniYSer6dpns0xdCq3kIEY\nESF2rh7cpF991eWa9M8MEv08vBREXn/vnJBbOIlriHmJ14cnT8gtjMVsOw8DMfK9oOVOuCE6eTIQ\njerDBB0dlpZtRfuonh5k3ngDEMkliJN1Kmkjs4cy9corVlXHUrncwokTfZ1bGMTrXO/MSW8G8U7l\nFjIQIyJIPK73FijlyW+n6R07gGwWkfHjEaqtdbs6lCc2dQoQDiPzxpuezPVJOzDrjcpj5ux5deak\nWS+781IZiJHvBS13wi29PRvWfju1on16v5n6N8fHyyppI6mpQfSss4xcH+9N9gjKeyeI17no1ClA\nKITMG29A9fS4XZ2TmEE8e8SIyBG5XJ/XvLfAYsqok5+TrYMsar53LA7irZCb5OHzQCyIQrW1iIwf\nD2SznlthX+vuRuaNNx3JLWQgRr4XxNwJN0SNIMfqm6kV7ROEdaC8rNI2itnUm1op1dOD9BtvAKFQ\nbjsmvwrqdc4cnkx7bHgys32HkVs4wfbcQgZiRAQA+vBSLIbM3r3Q2tvdrk6OSiaR3rkTANeB8qqo\nR2dOpnfuBDIZ5hZ6mNnLnfJYbmquF96Baw4DMfK9IOZOuEFisdzWQVZ+O620fdK7dgHpNCLNzQjV\n11tUK8pXaRtFJ03SZ93u2QPt+HGLalW53LDk2f6faRvU61xuCQuP9aaaXyqcWLeQgRgR5eSGmDzU\ns2FeoKPMD/MsiccRney9WbeprVsBALHp012uCfUnOm0aEAohvWMHtO5ut6uTk9q8BQAQm2H/e4eB\nGPleUHMn3NCbdG3dmlCVtk9q82YAQGzmTCuqQwVY8RmKnnMuAG/lifXeTGe4XJPKBfU6F6qrnrDB\nigAAHIFJREFUQ+Sss/SEfY98AdTa25HZvRuIxXLbv9mJgRgR5cTMm6lHLohA3s10pv9vpkHmtRX2\ntbY2ZN58U7+Z+jxRP+jML1mpLVtcrokutfUVfUX9qVMh8bjt52MgRr4X1NwJN0QmToAkEsgeeAvZ\nlhZLyqykfbS2Nke/mVYrKz5DXlth36yHUzdTuwX5OhebqQ//pTZvdbkmuvQWZ7/8MRAjohyJRHLD\nk6lNm12uTW+OT/TsswNxMw2y6Fn6FkLZ/QeQPXrU7erk9aRySNvrzKHj9Gb3rzlAb8+cE/lhAAMx\nCoCg5k64JTZ7NgAgtWmTJeVV0j4clnSGFZ8hiUYRNZLirXrvVCK9xUjUnxmMRP0gX+ciEyZAhgxB\n9t13kT10yNW6KKUcD+IZiBHRCWKzjHyNje7fTHsT9RmI+YFX3jv6zZSTPPxCQiHEzjXyU13OE8u+\n/Ta0o0cRampCeMwYR87JQIx8L8i5E26IzZoFAEhv3QqVyVRcXrnt48Y302pl1Wco15u6caMl5ZUr\n+9Zb0FpaEBo+HOHTT3e1LlYJ+nUuOsPME3M3EDMDweiMGRARR87JQIyIThAeMQLhM8+A6u7OrWjv\nhuy+/VCtrQidcgrCH/yga/Wg4uV6xF55BSqddq0e+T2pTt1MqTJemTnp5PphJgZi5HtBzp1wS2yW\ndT0b5bYPb6bOseozFG5qQmTcOKAnifS2bZaUWQ5zokkQ1g8zBf06Zwbx6S1boVIp1+qR2rDBqM8s\nx87JQIyITpLr2XBx5mTyT38y6uLcBZEqF5utt5ebeWLJl9frdbngfNfqQKUJDx+OyLhx+kbtLm0A\nrnV2Iv36NiAczl0DncBAjHwv6LkTbojncn0qv5mW2z6p9XogFr/wworrQAOz8jPkdp6Y1tqKzM6d\nQCwWqK2NquE6ZwbO5pcwp6U2bQKyWUSnnY1QXZ1j52UgRkQniUw6C1JXh+xbbyH73nuOnz975Agy\nu3dDampymwKTP5g9YskN7gRiyQ0bAaUQm34uJJFwpQ5UnvgFFwAAUuvXu3L+3Je/853tSWUgRr4X\n9NwJN0g43HtDfbmyi2I57WNeEGOzZkFisYrOT4Oz8jMUGT8e0tgA7dAhZN5+27Jyi5Uyh7Qdvpna\nrRquc7kesQ0boDTN8fMnzevOhRc4el4GYkRUUPyiiwAAqZdecvzcyfXM8fErCYVyPRvJdescP7/5\nxSHu8M2UKhf+4AcRHjUKqrUNmV27HD23SiaR2mJM8jiPPWJEJamG3Ak3xC++GACQfOnlisopp31S\n5s30At5MnWD1Zyj33lnrbBCvdXXpm46HQrlctaCohuuciOR6o8zeKaekXn0N6EkiMnEiwk3DHD03\nAzEiKih6zjRIbS0yu3c7miemtbUhvX07EI0iyhX1fSm/N1Up5dh505u3AJkMolOnIjRkiGPnJeuY\n+VkphxP23RzSrrpATETGishct+tB1qmG3Ak3SDSK2PnnAQCSL5ffK1Zq+6Q2btKTrc89F6GamrLP\nS8Wz+jMUmTwJ0tiI7DvvIHvggKVlDyTIQ9rVcp3L5YmtX+9oEJ8b0nbhvRPYQExEZorIYwWemgXg\nMRHRRKRFRFaJCL92ExVg9mwk11U2PFmKpJGT5nTCLFlHQiHEL9KXHal0aLsUybVrATA/zM8iEyYg\nNGIEtEPvIbN7tyPnVKkUUsaXTfN966TABWIiMkNElgC4EUBzgUOUUqoJQKNSqkkpdY1Syt09Fagi\n1ZA74RYz16eShP1S2yf5wh/0c196adnnpNLY8RnK5Yk5lLCvdXToCxCHw7lzB0m1XOckFEL8Mv2z\nn/zDi46cM7V5M1RXFyJnTUR41ChHzpkvcIGYUmqLUmoxgEcHOa7doSoR+VZ02tmQujpk9uxB9tAh\n28+Xff99pLdvhyQSiJ8XrGTrapPrEVu3zpEhpuS6l4BMBrHp0xFqaLD9fGSf+GWXAXAuEMt9+TPO\n67TABWJ5uDldlaiW3Ak3SDSKmDFzsafMi2Ip7ZN88Y8AgNhFF3IxTgfZ8RmKnHUWQk1N+hDTnj2W\nl99X8g/GzfRDl9t+LjdU03UucbkRiK1b58i+kz0vvmic1533TpADsX6JyJy8n4Uiwq9PRP1IXHUl\nACD5/PO2n6vn9y/o53TpgkjWkVAI8Ss+BABIPmf/e8fsPYnzveN74VGjEJk4EaqrC6nN9u53q7W2\nIv3Kq0A0ipgL+WFAdQZimwHsVUo9q5R6FsDjAFa4XCeqQLXkTrglceUVAICeF/4AlcmU/Ppi20dp\nGpLGN9Og9mp4lV2fITOI77E5EMv8+c/I7NkDGTIEsRnB2V8yX7Vd5+Jmr5gxbGiX5Np1gKYhNnsW\nQrW1tp6rPxFXzuoipdS+vo9FpFlEZvSXtL9w4ULUGRuATp8+HZdccknuQ2F2F/MxHwf2cV0dwmPH\nIrtvH95/7nmEZky35XyZHTuhHT4MjByJyMSJ3vn383HZjzvOPhsQQfLll6F1duJoV5ct56s1h81n\nz8aRY8c88+/n4/IfJy6/HJ0P/QQdzz2PoYvutO18md/9DoDeC19peeUSJ9fpKIWI3ALghiIPv0Ep\n1dbn9TMBPKiUGjTjV0Q2AlimlHqowHPKq/9HpDt8+HDVfVt0WuvXv4HOn/wE9V++HQ2LF5X02mLb\n5/iPl6H9W3ej9qYbMew73y63qlQGOz9Dhz/6caQ2bULTTx9CzbXX2nKOlgW3ofu3v0XDv9+N+r/5\nnC3ncFu1Xee0ri68O+VsIJPBaa9uRbipyfJzKKXw3gUXIXvwIE558n8Rm15Rb2rZeemeHZpUSi03\nlpYo5qdt8BIBo+er0E6iLcYPERWQuOoKAPbm+vSsWQOAw5JBEzeHJ5+1572jkkn0/P73AHqH0cn/\nQrW1+sxbpdCz5llbzpHeth3ZgwcRGjkS0XPOseUcxfBsIGaTowAWFPj9bOi5Y+RD1fQt0S3xC/VZ\njOlt20pexqKY9skePYrUnzYA0SgSV15ZbjWpTHZ+hhJzrgIAJJ97zpZlLJLr1kF1dCA6ZQoiY8ZY\nXr5XVON1LmH0oPasWmVL+T2rV+vnuXouJOReOBTkQOykfsxCPWciMh/Ao0qp/U5UisiPJJFA7JJL\nAAA9q9dYXn7PmjWApiF+ycUIDR1qefnknujUqQiNHInsu+8is32H5eX3/E6/SSeuvcbyssldNdfo\nbZr8/QtQ3d2Wl2++d8zzuCVwgZixl+QSAEsAzBCRZUa+GQB9yNNYsuIWEVkIYKhS6jbXKkwVq6b1\nddxU8xfzAADdTz1V0uuKaZ+ep5/Rz2FTDhENzM7PkIRCSFx9NQCg+8knLS1baRq6jd6SxLxgv3eq\n8ToXHj0K0XOmQXV3o+ePay0tO3PwHaRfew1SW4v4pZdYWnapAheIKaX2KaUWK6VmK6XCSqlblVLL\n+xxzn5GDdp9S6n636krkJzXXXgtEIkiuXYdsi3UplVpXV++CitdcbVm55B01H/lLAED3b5+0dHgy\nvWUrtPfeR/gDH0B06lTLyiXvSBi9VT1PP21puT3P6F/+4ld8yPXFowMXiFH1qcbcCTeEhg3Tvzlm\ns7ku/WIM1j49q1YBPUlEZ85E+LTTKq0mlcHuz1D84osQampCZs8eZHbstKzcrt/8BgCQmDcPIsHe\nTKVar3M1f/URAED3089A9fRYVm7Xr/X3Ts1HPmJZmeViIEZERav5S7Nn47eWldm18lcAgNpPfsKy\nMslbJBJB4sMfBmDde0dlMuj+zf8AAGo/+XFLyiTviY4fj+jZZ0O1t6PHot09MgcOIL15M6S21hO9\n8AzEyPeqMXfCLYl584BoFMk/vFj07MmB2id79CiSL7wAhMOo+ehfWVVNKpETnyGzZ6Pr17+G0gqt\nIlSa5Nq10I4cQXjsWETPPbfi8ryumq9zNZ/4GACg61e/saS8rl/9GoCeVxiqqbGkzEowECOiooWb\nhumJ15qGrpVPVFxe9//8L5DNIv6hDyE8fLgFNSSvil98EcKjRyN74C2kXnq54vK6ntBvprWf/ETg\nhyWrXe1HPwaIoGfNGmjt7RWVpZRCtzEsWftxb/SkMhAj36vW3Am31N10IwCg65ePFpV43V/7KKXQ\n9ehjADi05DYnPkMSDqPWeO90/vKXFZWltbWhx5iB6ZWbqd2q+ToXHj0K8YsvBpJJdD1R2RfA1IYN\nyLz5JkIjRuT2s3QbAzEiKkn8ig8hdOpIZPbuRWrDhrLLSW/eok8fb2xEzbx5FtaQvKr2Rn3Xuu6n\nnoLWVtSGKAV1Pb4Sqrsb8UsvRaR5rFXVIw+rvfkzAIDOh/+7opm3nQ//t17epz8FiUYtqVulGIiR\n71Vz7oQbJBJB7Q36DbXzp/816PH9tU/Hz34OAKj7609DPJCnUc2c+gxFxoxB/NJLgZ5krje0VEop\ndP78YQBAXUD3lSyk2q9zNfOuRWjkSGR2vYHU+vVllZE9ehTdTz4FiKDOCOy8gIEYEZWs/m//BohE\n0P3kU8i89VbJr88eOaLPnhNB3ec+a0MNyavq/u4LAICO5Q9BpdMlvz754ovI7NmD0GmneWLGGzlD\nYjHU/fWnAQAdP/1ZWWV0/b9fAKkUEnPmIPLBD1pYu8owECPfq+bcCbeER41Czcc/DmgaOpY/NOCx\nhdqn44EH9Qvi1XMROf10u6pJRXLyM5SYOweRCROQfeed3PITxVJK4fj3/wMAUP+5z0IiETuq6Em8\nzkHvxYrF0PPUU0jv2lXSa7WODnQ8qK/tbn4Z8AoGYkRUliG3zgcAdD3yy5I2As8ePozO//qZXsb/\n/Xs7qkYeJqEQ6m9bAAA4/sMfQWUyRb82tXYdUi+vhzQ2oO4Ln7eriuRR4VGjUPfpTwFK4fj3vl/S\nazt/9nNox44hNns24pddalMNy8NAjHyv2nMn3BKdPBmJv/gwVHc32pfe2+9xfdvn+H/+EKq7G4lr\nr0Fs+nS7q0lFcPozVPuJTyB8+unIvPEGun7xSFGvUUqh/T59R7ohCxYgNGSInVX0HF7ndENuvx2I\nxdD9v79FushN5LW2NnQse0B//R3/4LnlThiIEVHZGv7pLiAaRdeKx5F69dVBj0/v3IlOI0l/6B13\n2F098iiJxdDwta8CANrvvQ9aa+ugr+l+4ldIbdyIUFMTe8OqWHj0KH2IUim0fvWrRc2gbL/vfr03\n7ILzEb/MG0tW5GMgRr7H3An3RMaORf0XPq9fFO9YWHAvOLN9VDqNY1+5E8hkUPe5zyI6dYrT1aV+\nuPEZSvzFhxG76CJox46h7Rv/OuCx2ffeQ9u//hsAYOhX/xmh+nonqugpvM71GnrHPyI0YgRS6/+U\nm0Hbn+T69fox4TAav/Utz/WGAQzEiKhCQ/7xHxA+8wykt29H69e+XvAbqlIKbd/8FtJbtiB02mkY\netdiF2pKXiIiaLznbkgiga4Vj6OznyFKlUyi5Ytfgnb0KOKXXZZbi4yqV6ixEQ3f1APztn/7JlJb\nthQ8LnvoEFq+eDugaai/7VZEp0x2sppFYyBGvsfcCXeF6uvR9MP/BBJxdP3iEbR941+hstnc8+8f\nOoT2JUvR+ZOfAtEomh5YhtDQoS7WmPpy6zMUnTABDXd/EwDQumgxOvusLaZ1dKBl/q1IvbweoZEj\nMewH3/dkj4YTeJ07Ue1H/wq1N98MJJM4cvNnkdy46YTnM2+/jSOf+mtohw4hNns2hn7Fu6kQUskK\ntdVARBT/j7zt8OHD7Lb3gO5nnkHLgtuATAbRadNQ+6mbABG0/+IRqNdfB8JhDPvBf6D2Yx91u6rU\nh9ufofZvfwfHv/NdAEDi6rlIzLsWWssxdD7838i+/TaksQGnPPZYVQ9nu91GXqTSabTMX4CeVauB\nSAS1N92E2OxZyOzZo6/A396OyFkTMeLxxxFuGmZ3dcr+hsBAbBAMxIiKl3zpJbR86XZo771/wu9D\np5yCYd/9NhJXXulSzcjLlFLoeuSXaPva10/KM4xMnoymH/8Q0QkTXKodeZlKpdD27/egs8B6holr\nrsaw73wboWG2B2EAAzH7MBAjKo3W3a3PcNuwARBBbOZM1Hz8Y1W33ACVLnv4MLpWPI709u2Qunok\nLrsUiXnXVtXCrVSe9O7d6Hp8JbIHDiB06qmo+fA8xM4/38mhbAZidmEg5n3ssvc2to/3sY28j23k\neWUHYkzWJyIiInIJe8QGwR4xIiIiGgR7xIiIiIj8hoEY+R7X1/E2to/3sY28j20UXAzEiIiIiFzC\nHLFBMEeMiIiIBsEcMSIiIiK/YSBGvsfcCW9j+3gf28j72EbBxUCMiIiIyCXMERsEc8SIiIhoEGXn\niAVyAy8RucX46yzjz0VKqba85+cDOGo8bFZK3edk/YiIiIiAAA5NisgtSqnlxs+tADYZP+bz8wFo\nSqmVSqmVANaIyDK36kuVY+6Et7F9vI9t5H1so+AKVCAmIg19f6eUWg6gSUSuMn41Xyn1UN7zWwDM\ndaiKZIO1a9e6XQUaANvH+9hG3sc28jYRuaLc1wYqEAMwDsADIjK0z+/3AmgWkUYAMwu8rlVE5the\nO7LF1q1b3a4CDYDt431sI+9jG3neFeW+MFCBmFJqM4CZSqn2Pk81wwjGALQWeGkLCgdoRERERLYJ\nVCAGAEqpE742iMj1APYopZ4D0AQ96OqrFcBwB6pHNujs7HS7CjQAto/3sY28j20UXIFevsIYilwD\n4CqlVLuIzAWwTCk1vs9xj0EP1u4qUEZw/4OIiIjIEkqpspaw8OzyFcYSFDcUefgN+ctT5FkC4Po+\nQ5VNBY5rRO9yFico9z+WiIiIaDCB7RETkYUAViil9uf9rhFAi1Iq1OfYjQDuNIYviYiIiBwRuBwx\nINeb1jcIm6OUagWwt8AyF40MwoiIiMhpgQvEjDywjWYQJiKNxu9MSwHclXf8TACrHa0kEREREQI2\nNCkizQB2F3hKARhm5ooZPWZ7oeeGFdziiNsged9gW1mRd4jICqVUsTmf5CAjjaMV+oxyUUo97nKV\nKE/eda4R+uz+e3idc4/RebNYKXVjgefKihsCFYhZJW8bpIeMxzMALDC2TCIPMLeyyn8MPRAbP8DL\nyAXGhWtj39xMcp8xY/zOvBEEDXqqRt+1GMkFRpD8QH57iMhjhYIAspcRB9xkPJyrlJrd5/my4wZe\nGAvjNkgeNshWVtwhwXsKzVQmlxk3jj/l59JC/xbPIMw7zivQHoXynMlmSqktSqnFAB7t55Cy4wYG\nYn1wGyRfGGgrq7Eu1If6ISLXKaXWuF0PKmgJgBOGIfsEZeS+ZqNnJV8jhyZdddKSVpXGDQzETsZt\nkDyuiK2syAOMG8gmt+tBJzNuHI36X+U6EZkjIgvZ0+I5twDYZAxRwripL3O3SlRARXEDA7GTcRsk\nHxhkKyvyhmb2sHiWeeNoUEqtVEo9C+BBAM+6Wy3KZwxvjQNwl4i0GL/j7t/eU1HcwECsNJzZ4EHG\nt/vFADh07BHGkORKt+tB/WqC3iOW60E2h7uYguEdxkoA1wM4E3qgvDpvFiX5w6BxAwOxwkraBolc\nV2grK3KJiIwFh4i9bi8AFPjMMAXDW+5USt2nlGo3EsVnAVjKYNmTyo4bPLvXpIs2Qv/P66sJwGaH\n60KDMHInlnAIzFPmAui7kHJuvar8ZUfIHUqpvSL9bqN7zMm6UGFGsLUq/3dKqS0icgOAq8FhZC+p\nKG5gINaHUqpVRPaKSEOfmSncBsljBtjKihcoFxUKtERkKRdF9pzNIjJWKbUv73fN0G8q5A2FouV9\n4OiMp1QaN3BosjBug+RxRWxlRUQDW2T8AMhd5/YwGdwbjC+UNxV46joADzhcHerV37qIZccNXFm/\nH8Vsg0TuKHYrK3KfMbyyAPrNYyX0VcLZY+kRInId9F4wABhu5CGRRxjLidwFvQesFfr9aAVTMZxn\n5L4ugJ56MQPAcgCbCuzwUnLcwECMiIiIyCUcmiQiIiJyCQMxIiIiIpcwECMiIiJyCQMxIiIiIpcw\nECMiIiJyCQMxIiIiIpcwECMizxCRuSKilfizu08Z14vIMWONLN8QkfnGulGlvGahXfUhImcwECMi\nLzH3a1PQV6puhr7RsWm1cczV6N3DbVifMuYDaABwo33VtJaIrAAwp8/2KMVoFZHdpQZwROQd3GuS\niLzE3D7kXqWUuV3IfhFpgx5cbTZ2TnjWWLV/H07ebHcB9GDMF9vAiMgDAM5USp1X6muVUstFZBaA\nTQDGW145IrIde8SIyIvuGewAo/foHvTZGFkptU8pdZcftoERkesB3IK8PR/LsAhAs4gss6ZWROQk\nBmJE5CWN0PdvK3a/UL/vW2nuV/dcuQUYAem9AOaLyAzLakZEjmAgRkRe0gRgY7EHK6U2A4CIDLWt\nRjYxesMaAKyxoLjVxp8LLCiLiBzEQIyIvGQ1Ss/tuhOAiMgDfWZTLjEPEJGlfZ67xZhducmceSki\ntxjHNovIChFpMX6W9HdiY6bjaqOMjSXOYjSDpg39lH29iOzJq99Co16F6mMGr7eUcH4i8gAGYkTk\nGUqpZ5VSW0t8zf3G8NydAMahdzalyjtmkfHcXuNXCwA8CGCV8WczgAdE5E7oie8agH8H0ALgzkL5\nV0aS/TIAjyqlQtBztZaKyGNFVn2OUcfNfZ8QkblGva4zyr7a+Lku/9+V9+9rA9BmvJbDk0Q+wkCM\niAJBKdWmlNqHfvLGjOfMoGcGgJlGUv+t0HOsAGAJgGVKqZuUUvdDD34A4Jb8JSLykuxXK6UeMsp/\n1ijnemNGZ79EJH+mZ0uBQ26AHuBtNeuulLoGvYFkIS3QJy40D3RuIvIWBmJEVE3MGZaP95lVaeZY\nKeTN2DSCN9PYvL8vNf7sO4y6us/z/TGX6UA/ExOaANxUYFHaRQCO9lNma4F6EpHHMRAjomrUNy/r\nmPFna4HAaB9ONhZ60Nb3OXNx2ZmVVQ+roM8gXWHkiK0y8s82Gz11RBQQDMSIqBq19vP7QsOEJxAR\nc+hPAJjJ/pqIaADM/DA1yEzO3HkKHaeUWg59mFMZP3Oh97LtMScVFGAOdw40fElEHsNAjIioNGYQ\npQA0KqVCBX7CA62FppRqhZFcD2B43+dFZK5SarFSKgzgGuhBmRk89jertMmoEwMxIh9hIEZEVAIj\niDIV3JZIRIoZmtwAvVet0LErzIR/YybpYqVUE4wV+EXkzD7na4S+JhlKnXVKRO5iIEZEVLp7oQdR\nJ21NZCw9UcwSFmbPVn97TBZK+H8WAAps3zTb+PPBIs5LRB7CQIyIPEtEGo3Axlw6YpaIjO2z/ENf\nZg7XuAGe67tBtvn7k4YJoQ/5CfICJqXUYuhLYcw1Eunni8hcEVkKPQi7foD6mWWshD7c2N+xM4yy\nZwC53LTlKDw0aS6zMdhsTSLyGFHqpLUBiYhcZyyuaq4in3+hEgDKyJ/KP97siWrIO64V+ppcw4zn\nVN5zx6AHYPuN1+Q/Z/Z0Le3z+z1KqQl551wI4Cbow4ut0JevWFTshuPG8hQrAFxtrENm/v4o9AVf\nz4O++OxM6LlfK5RSd/UpoxF63toDSqnbijkvEXkHAzEiIhcZq/bPVUr17aUr9vUPALgqP0AkIv/g\n0CQRkYuMlf3XlLA1Uo6IzAdwFYBZlleMiBzBQIyIyGVGMLY6fxul4l+qJgy0VAYReRuHJomIiIhc\nwh4xIiIiIpcwECMiIiJyCQMxIiIiIpcwECMiIiJyCQMxIiIiIpcwECMiIiJyCQMxIiIiIpcwECMi\nIiJyyf8HlFmUyTHV5KkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112494390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set the plot size - 3x2 aspect ratio is best\n",
    "fig = plt.figure(figsize=(6, 4))\n",
    "ax = plt.gca()\n",
    "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n",
    "\n",
    "# Change the axis units to serif\n",
    "plt.setp(ax.get_ymajorticklabels(), family='serif', fontsize=18)\n",
    "plt.setp(ax.get_xmajorticklabels(), family='serif', fontsize=18)\n",
    "\n",
    "# Remove top and right axes border\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "\n",
    "# Only show axes ticks on the bottom and left axes\n",
    "ax.xaxis.set_ticks_position('bottom')\n",
    "ax.yaxis.set_ticks_position('left')\n",
    "\n",
    "# Turn on the plot grid and set appropriate linestyle and color\n",
    "ax.grid(True,linestyle=':', color='0.75')\n",
    "ax.set_axisbelow(True)\n",
    "\n",
    "# Define the X and Y axis labels\n",
    "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n",
    "plt.ylabel('Angle (deg)', family='serif', fontsize=22, weight='bold', labelpad=10)\n",
    "\n",
    "# Plot the data\n",
    "plt.plot(sim_time, response[:, 1] * 180/np.pi, linewidth=2, linestyle='-', label = '$\\theta$')\n",
    "\n",
    "# uncomment below and set limits if needed\n",
    "# plt.xlim(0, 5)\n",
    "# plt.ylim(-1, 1)\n",
    "\n",
    "# Create the legend, then fix the fontsize\n",
    "# leg = plt.legend(loc='upper right', ncol = 1, fancybox=True)\n",
    "# ltext  = leg.get_texts()\n",
    "# plt.setp(ltext, family='serif', fontsize=20)\n",
    "\n",
    "# Adjust the page layout filling the page using the new tight_layout command\n",
    "plt.tight_layout(pad=0.5)\n",
    "\n",
    "# Uncomment to save the figure as a high-res pdf in the current folder\n",
    "# It's saved at the original 6x4 size\n",
    "# plt.savefig('Payload_Response.pdf')\n",
    "\n",
    "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also plot the horizontal position of the trolley and payload over time, to better understand how they interact. To do so, we need to define the horizontal position of the payload, $x_p$. It is simply the position of the trolley, $x$, plus the horizontal position of the point mass relative to it, $l\\sin{\\theta}$:\n",
    "\n",
    "$$ x_p = x + l\\sin{\\theta} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the horizontal position of the payload\n",
    "xp = response[:,0] + 2.0 * np.sin(response[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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28e7h+L8mWc2AVxit9VNa6+dG+wFa6w3hOI4QoXh97xkAls4ZT3JS9EbWBXPb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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1124b1dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Now plot the responses together\n",
    "\n",
    "# Set the plot size - 3x2 aspect ratio is best\n",
    "fig = plt.figure(figsize=(6, 4))\n",
    "ax = plt.gca()\n",
    "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n",
    "\n",
    "# Change the axis units to serif\n",
    "plt.setp(ax.get_ymajorticklabels(), family='serif', fontsize=18)\n",
    "plt.setp(ax.get_xmajorticklabels(), family='serif', fontsize=18)\n",
    "\n",
    "# Remove top and right axes border\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "\n",
    "# Only show axes ticks on the bottom and left axes\n",
    "ax.xaxis.set_ticks_position('bottom')\n",
    "ax.yaxis.set_ticks_position('left')\n",
    "\n",
    "# Turn on the plot grid and set appropriate linestyle and color\n",
    "ax.grid(True,linestyle=':', color='0.75')\n",
    "ax.set_axisbelow(True)\n",
    "\n",
    "# Define the X and Y axis labels\n",
    "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n",
    "plt.ylabel('Position (m)', family='serif', fontsize=22, weight='bold', labelpad=10)\n",
    "\n",
    "# Plot the data\n",
    "plt.plot(sim_time, response[:, 0], linewidth=2, linestyle='--', label = 'Trolley')\n",
    "plt.plot(sim_time, xp, linewidth=2, linestyle='-', label = 'Payload')\n",
    "\n",
    "# uncomment below and set limits if needed\n",
    "# plt.xlim(0, 5)\n",
    "plt.ylim(0, 2)\n",
    "\n",
    "# Create the legend, then fix the fontsize\n",
    "leg = plt.legend(loc='upper right', ncol = 2, fancybox=True)\n",
    "ltext  = leg.get_texts()\n",
    "plt.setp(ltext, family='serif', fontsize=20)\n",
    "\n",
    "# Adjust the page layout filling the page using the new tight_layout command\n",
    "plt.tight_layout(pad=0.5)\n",
    "\n",
    "# Uncomment to save the figure as a high-res pdf in the current folder\n",
    "# It's saved at the original 6x4 size\n",
    "# plt.savefig('TrolleyAndPayload_Response.pdf')\n",
    "\n",
    "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given our choice of masses, we see that the payload is able to back-drive the trolley, causing the trolley to move opposite of the payload once the bang-bang command is over."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr class = \"style-end\">"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Licenses\n",
    "Code is licensed under a 3-clause BSD style license. See the licenses/LICENSE.md file.\n",
    "\n",
    "Other content is provided under a [Creative Commons Attribution-NonCommercial 4.0 International License](http://creativecommons.org/licenses/by-nc/4.0/), CC-BY-NC 4.0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Ignore this cell - We just update the CSS to make the notebook look a little bit better and easier to read\n",
    "\n",
    "# Improve the notebook styling -- Run this first\n",
    "import codecs\n",
    "from IPython.core.display import HTML\n",
    "css_file = 'styling/CRAWLAB_IPythonNotebook.css'\n",
    "HTML(codecs.open(css_file, 'r', 'utf-8').read())"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}