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      "The Extended Kalman Filter"
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     "source": [
      "**Author's note: this is still being heavily edited and added to - there is a lot of duplicate material, incorrect code, and so on. The examples are just a brain dump, not fully formulated and not ready to be used.**\n",
      "\n",
      "**The Uncented Kalman filter (UKF) chapter is much further along. The UKF is almost always better performing than the Extended Kalman filter, and is much easier to implement, so if you have an urgenet need for nonlinear Kalman filter I'll point you towards that chapter for now.**\n",
      "\n",
      "The Kalman filter that we have developed to this point is extremely good, but it is also limited. Its derivation is in the linear space, and hence it only works for linear problems. Let's be a bit more rigorous here. You can, and we have in this book, apply the Kalman filter to nonlinear problems. For example, in the g-h filter chapter we explored using a g-h filter in a problem with constant acceleration. It 'worked', in that it remained numerically stable and the filtered output did track the input, but there was always a lag. It is easy to prove that there will always be a lag when $\\mathbf{\\ddot{x}}>0$. The filter no longer produces an optimal result. If we make our time step arbitrarily small we can still handle many problems, but typically we are using Kalman filters with physical sensors and solving real-time problems. Either fast enough sensors do not exist, are prohibitively expensive, or the computation time required is excessive. It is not a workable solution.\n",
      "\n",
      "The early adopters of Kalman filters were the radar people, and this fact was not lost on them. Radar is inherently nonlinear. Radars measure the slant range to an object, and we are typically interested in the aircraft's position over the ground. We invoke Pythagoras and get the nonlinear equation:\n",
      "$$x=\\sqrt{slant^2 - altitude^2}$$\n",
      "\n",
      "So shortly after the Kalman filter was enthusiastically taken up by the radar industry people began working on how to extend the Kalman filter into nonlinear problems. It is still an area of ongoing research, and in the Unscented Kalman filter chapter we will implement a powerful, recent result of that research. But in this chapter we will cover the most common form, the Extended Kalman filter, or EKF. Today, most real world \"Kalman filters\" are actually EKFs. The Kalman filter in your car's and phone's GPS is almost certainly an EKF, for example. \n",
      "\n",
      "With that said, there are new techniques that have been developed that both perform equal to or better than the EKF, and require much less math. The next chapter "
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     "source": [
      "The Problem with Nonlinearity"
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     "source": [
      "You may not realize it, but the only math you really know how to do is linear math. Equations of the form \n",
      "$$ A\\mathbf{x}=\\mathbf{b}$$.\n",
      "\n",
      "That may strike you as hyperbole. After all, in this book we have integrated a polynomial to get distance from velocity and time:\n",
      " We know how to integrate a polynomial, for example, and so we are able to find the closed form equation for distance given velocity and time:\n",
      "$$\\int{(vt+v_0)}\\,dt = \\frac{a}{2}t^2+v_0t+d_0$$\n",
      "\n",
      "That's nonlinear. But it is also a very special form. You spent a lot of time, probably at least a year, learning how to integrate various terms, and you still can not integrate some arbitrary equation - no one can. We don't know how. If you took freshman Physics you perhaps remember homework involving sliding frictionless blocks on a plane and other toy problems. At the end of the course you were almost entirely unequipped to solve real world problems because the real world is nonlinear, and you were taught linear, closed forms of equations. It made the math tractable, but mostly useless. \n",
      "\n",
      "The mathematics of the Kalman filter is beautiful in part due to the Gaussian equation being so special. It is nonlinear, but when we add and multipy it using linear algebra we get another Gaussian equation as a result. That is very rare. $\\sin{x}*\\sin{y}$ does not yield a $\\sin(\\cdot)$ as an output.\n",
      "\n",
      "> If you are not well versed in signals and systems there is a perhaps startling fact that you should be aware of. A linear system is defined as a system whose output is linearly proportional to the sum of all its inputs. A consequence of this is that to be linear if the input is zero than the output must also be zero. Consider an audio amp - if a sing into a microphone, and you start talking, the output should be the sum of our voices (input) scaled by the amplifier gain. But if amplifier outputs a nonzero signal for a zero input the additive relationship no longer holds. This is because you can say $amp(roger) = amp(roger + 0)$ This clearly should give the same output, but if amp(0) is nonzero, then\n",
      "\n",
      "> $$\n",
      "\\begin{aligned}\n",
      "amp(roger) &= amp(roger + 0) \\\\\n",
      "&= amp(roger) + amp(0) \\\\\n",
      "&= amp(roger) + non\\_zero\\_value\n",
      "\\end{aligned}\n",
      "$$\n",
      "\n",
      ">which is clearly nonsense. Hence, an apparently linear equation such as\n",
      "$$L(f(t)) = f(t) + 1$$\n",
      "\n",
      ">is not linear because $L(0) = 1$! Be careful!"
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     "source": [
      "The Effect of Nonlinear Transfer Functions on Gaussians"
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     "source": [
      "Unfortunately Gaussians are not closed under an arbitrary nonlinear function. Recall the equations of the Kalman filter - at each step of its evolution we do things like pass the covariances through our process function to get the new covariance at time $k$. Our process function was always linear, so the output was always another Gaussian.  Let's look at that on a graph. I will take an arbitrary Gaussian and pass it through the function $f(x) = 2x + 1$ and plot the result. We know how to do this analytically, but lets do this with sampling. I will generate 500,000 points on the Gaussian curve, pass it through the function, and then plot the results. I will do it this way because the next example will be nonlinear, and we will have no way to compute this analytically."
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      "import numpy as np\n",
      "from numpy.random import normal\n",
      "\n",
      "data = normal(loc=0.0, scale=1, size=500000)\n",
      "ys = 2*data + 1\n",
      "\n",
      "plt.hist(ys,1000)\n",
      "plt.show()"
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9Wul0uqBNW1ubFi9erI6ODklSR0eHZs6cqeXLl+fbtLe3q76+Pt8GAIZj2/aw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\nK/1/Ahe9hrqv07kAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914ad4bcf8>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is an unsuprising result. The result of passing the Gaussian through $f(x)=2x+1$ is another Gaussian centered around 1. Let's look at the input, transfer function, and output at once."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from nonlinear_plots import plot_transfer_func\n",
      "\n",
      "def g(x):\n",
      "    return 2*x+1\n",
      "\n",
      "plot_transfer_func (data, g, lims=(-10,10), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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A8+gdHC1PYtRqsSAhg3KwdUJQh+4AoP1vY2QVpKG65uYusCUVhfjtj591lkXO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\nqqpUVlamZ5991nSUoDV79mxNnz79nCJXXFyspUuX6umnnza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       "text": [
        "<matplotlib.figure.Figure at 0x7f914ad4bc18>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The plot labelled 'input' is the histogram of the original data. This is passed through the transfer function $f(x)=2x+1$ which is displayed in the chart to the upper right. The red lines shows how one value, $x=0$ is passed through the function. Each value from input is passed through in the same way to the output function on the left. The output looks like a Gaussian, and is in fact a Gaussian. We can see that it is altered -the variance in the output is larger than the variance in the input, and the mean has been shifted from 0 to 1, which is what we would expect given the transfer function $f(x)=2x+1$ The $2x$ affects the variance, and the $+1$ shifts the mean.\n",
      "\n",
      "Now let's look at a nonlinear function and see how it affects the probability distribution."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from nonlinear_plots import plot_transfer_func\n",
      "\n",
      "def g(x):\n",
      "    return (np.cos(4*(x/2+0.7)))*np.sin(0.3*x)-1.6*x\n",
      "\n",
      "plot_transfer_func (data, g, lims=(-4,4), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ghJRzGOE3CcZGpi1qw8rc5mHDIyIDxzxj+JztXLHsDx/g+IV9+PXUZlRU3dWq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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914bb762e8>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This result may be somewhat suprising to you. The transfer function looks \"fairly\" linear - it is pretty close to a straight line, but the probability distribution of the output is completely different from a Gaussian.  Recall the equations for multiplying two univariate Gaussians:\n",
      "$$\\begin{aligned}\n",
      "\\mu =\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}\\mbox{, } \n",
      "\\sigma = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "These equations do not hold for non-Gaussians, and certainly do not hold for the probability distribution shown in the 'output' chart above. \n",
      "\n",
      "Think of what this implies for the Kalman filter algorithm of the previous chapter. All of the equations assume that a Gaussian passed through the process function results in another Gaussian. If this is not true then all of the assumptions and guarantees of the Kalman filter do not hold. Let's look at what happens when we pass the output back through the function again, simulating the next step time step of the Kalman filter."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y=g(data)\n",
      "plot_transfer_func (y, g, lims=(-4,4), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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TeV9NJSfArIZgKjmBhp9TJb+CsXDhQmzZsqXWkz6RnIJ9B1drKyzOx7aob1Bw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\nbyKlAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f9167bfa160>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As you can see the probability function is futher distorted from the original Gaussian. However, the graph is still somewhat symmetric around $0$, let's see what the mean is."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print ('input  mean, variance: %.4f, %.4f'% (np.average(data), np.std(data)**2))\n",
      "print ('output mean, variance: %.4f, %.4f'% (np.average(y), np.std(y)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "input  mean, variance: 0.0002, 1.0015\n",
        "output mean, variance: -0.0276, 2.2445\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's compare that to the linear function that passes through (-2,3) and (2,-3), which is very close to the nonlinear function we have plotted. Using the equation of a line we have\n",
      "$$m=\\frac{-3-3}{2-(-2)}=-1.5$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def h(x): return -1.5*x\n",
      "plot_transfer_func (data, h, lims=(-4,4), num_bins=300)\n",
      "out = h(data)\n",
      "print ('output mean, variance: %.4f, %.4f'% (np.average(out), np.std(out)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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i50NrcTzrACZFztY7jDsRGbZmvTRQXFwMjUbDq0qtVNSACQCApMx4ZGafqBO/\nVXITZibm8HL31y7r3MGXkywRkWRYZ4yXqYkpRg6YgL4+g7Bh3zKcv3ZaJ55bcBlLNr6HsN5RGD1w\nCizNrWTKlIiaShBFUXz0ao9n/PjxuHDhApKTk7UPbanVam383LlzzXVoMgCV1RUoKM3Rvr508zT8\n3ULgYO0iY1ZErY+3t7f2e5VKJWMmLU9fnQFYa4yNKIq4cOMkki/HorK6vE7cyswW/buORCcnXxmy\nI6Km1plm68F4++23cfjwYSQkJHBEiDbKzMQcbvZdta9Vlk5IvLATrnaej9zW3aEbGyJE1CDWmdZD\nEAR0cw2Eu4M3ki/vxaWbur0ZdypLEJ+5CR6Ovujf9WlYm9vVsyciMgTN0oMxb948/PDDD4iLi4OP\nj+69kw9eVTLWK23JyckAgKCgIJkzeXxyvAdRFFFVU/nI9W4X38Suoxvg4uDe4Hq5ubkAgP69n0Iv\nrxBJcmxp/F2Sn7HnD7SO82pTNVRnAOP5TIzp968lc83MPoGNcctQqM6vEzM3s0T0wMkI6xUFhUKp\nd3tj+VyNJU+AuTYHY8kTaPo5VfIejDlz5mDTpk31nvSp7RIEoVEPfLs6dsT0qHceuV7tf8y0C4dw\n7eYlnVhpeTFGhUzSmReEiFoH1pnWz88zEO9P/hq7j/2AfSlbtRPIAkBFZTm27F+J5Mz9mBg5G+7t\nusiYKRHpI2kD44033sC6deuwdetWqFQq5OXdG8va1tYW1tbWUh6KSGvKiDl1lmVdOYm4lG31DnF4\nu/gmJo/4Y3OnRkQSY51pO8xMzTHmqT8gyDcMG2KX4XJelk48O/8cvlj/JwzpG4OoARNhZsoRC4kM\nhaQNjGXLlkEQBERGRuosX7BgAT7++GMpD0XUIN9OveHbqXe98RPnDmPnQ/N+iKIIE6UJhgWNg7Ke\nbncikhfrTNvj5twZc19YhEOn9mD74W9RUXn/IXCNqEHs8Z9w4txhjB86E909+8iYKRHVkrSBodFo\npNwdUbMJ9A5FoHeozjKNqMHG2GVIPZsAN+fO2uXWlrZQWTu2cIZEpA/rTNukUCgR1nsUenoNwJb4\nFTh5IVEnXlicj2VbF6KfTxieDX9ZpiyJqBYnJCD6nUJQYPTAF6FUmiL/dg7yb+cg68pJxKdulzs1\nIiICYG/jhJej/4xXn/kA9jZOdeLHzx7Eom/fxLm8VDTjKPxE9AjNOtEekbGxs3ZAnwd6NhLTY1FY\nnIeDabsEqCwiAAAgAElEQVQea3+dXLrBs733o1ckIqJG69m1P7w79sSOI9/hwIkdEHG/MXGnohRH\nLuzAhZtp8PBqj/aOHjJmStQ2sYFB1IC+voPg37nfY24tYu3uJej6wEzmD6sdatfWxbTBZ0aIiEiX\nhZklxkW8giDfCGzY9w1yHhpN8EbxVfztu3kYHjQOw4PHwdTETKZMidoeNjCIGmBmYt6ooXXr8+a4\nTxqM1w61ezBtFy7kntEu12hqEB065bGPS0TUVni298Y7E79EfOrP2JW4HpXVFdpYjaYau49tRMrZ\ng5gQORveHXvImClR28EGBpEBeCX6zzqvD5zcoTPK1Z27pRgYMIzjvRMR6aFUKBHZbywCvQdi077l\nOJOdohO/UZSLf235ECH+kYgJmw5rC1uZMiVqG9jAIDJA4b1H67y+XngVSRlxdUZO0Se3IBvTRv4J\npiamzZUeEZFBcrJzxesxH2Hz7rVIuvQr7laV6cQTz8Qi/VIyng1/Cf18wyEIgkyZErVubGAQGYEO\nTh4YM2hqo9a9kJOOvUmbgQfqZnHZLXRu7wcfj17aZSobR873QUStjiAI6NIuAG4OXXG1LA2HT+/V\niZeUq7F2z1c4lhmP8UNeh7OqvUyZErVekg5Te+DAAYwZMwYdO3aEQqHAmjVrpNw9ETWCl3sARg2c\nhFEh97+iQ/8AESKyrp5E1tWT2Jn4PU5fTMKV/PNyp0vUJKwz1FjmJpaYGPkG5jz//+Dq2LFOPDM7\nFYvX/RF7k39ETU21DBkStV6S9mCUlZWhV69emDZtGqZOncquRyIDYWNph4EBw7Svu3v2Qf6ta4hL\n2YaALkFN2pezfQd0bu8jdYpEjcI6Q03l5R6Adyd9hdjjP2JP0iadxkRVdSV+PrQWx7MOYGLkbJ7b\niCQiaQMjKioKUVFRAIDp06dLuWsikpC9jRPsbZzQ3skDFZXlTdr2pwOr0NWtOwb1ioKluVUzZUik\nH+sMPQ5TE1OMHDABfX0GYcO+ZTh/7bROPLfgMr7a+B4G9YpCdOgUntuInhCfwSBqw1TWjoB107Z5\nafS7SDl7ELuPboC5maV2Lo8b1ef0ri9AQFTIxCdNlYjoibk4uOOt5z7B0TP7sDVhNe7cLdHGRIg4\nmLYTaReP4vmIV9G7W4iMmRIZN0EURfHRqzWdra0tli5diqlTdR9MVavV2u/PndP/BwkRtR7pOYmo\nqvl9XHpRhIO1Kzydu8ubVCvj7X1/tniVSiVjJi2rvjoDsNbQo92tKkPypb24ePO03riHoy/6d30a\n1uZ2LZwZkeFpap1hDwYRNasA9/tXAWs01Ui+tBe379xocBsThSl6dAxt7tSIqA2zMLXGIJ+x6OrS\nC4kXdqL0bpFO/OqtLOSpLyGw0xD4dugHhSDpuDhErZqsDYygoKY9XGooamdfNtb8Ab4HQ9EW38OA\n/o++7WDPsR9wo/ocqqorobJ2xAD/oTpxUxMzmCilmeejNfwMHrxaT3UZ8s/WmH7/Wm+uQRgZEYPd\nx37AvpSt0GhqtJGqmkokXdqDG3cuYmLkbMknO229n6m8jCVXY8kTaHqdYQ8GERmcp/uPBwBUVlfg\nUNoeJKbHamMl5WpYmlkhoEs/vdtamFnB0c6lRfIkotbBzNQcY576A4J8w7Ahdhku52XpxLPzz+GL\n9X/CkL4xiBowEWam5jJlSmQcJB+mtvZeV41Gg+zsbJw4cQJOTk7w8PCQ8lBE1AaYmZhjSN8xOsvu\nVpYjMzsVN27n6t3mxPkjGBgwDL6derdEitTCWGeoObk5d8bcFxbh0Kk9+PnwOtytvKONaUQNYo//\nhBPnDmP80Jno7tlHxkyJDJukDYykpCQMHXrvVgZBEDB//nzMnz8f06dPx3//+18pD0VEbZSFmSUC\nvet/PsPFwR37Urainb0bHO3atWBm1BJYZ6i5KRRKhPUehZ5eA7AlfgVOXkjUiRcW52PZ1oXo5xuO\n58Jfgq2VvUyZEhkuSRsYgwcPhkajkXKXRERN4ubsiQH+kfjlyDo4q9prl3ds1xW9vAbImBlJgXWG\nWoq9jRNejv4zTl08hk1xy1FUWqgTP551ABmXUxAzaBpCAoZx0keiB/AZDCJqdbw79oB3xx46y9bu\n+QrXbl6ss+6D83jU1FQjoEswurr5tUieRGT4enbtD++OPbHjyHc4cHInRPF+A/dORSnWxy7Fscx4\nTBw6C66OHWXMlMhwsIFBRG3C1Kfn6V3+4Cget4pv4rfjPyLzSqrOOuUVZRgX8Uqz50hEhsnCzBLj\nIl5BkG8ENuz7Bjk3L+nEL+Sk47Pv52J40DgMD3oepibSjHJHZKzYwCAi+p2jXTuMH/K6zrKy8mJs\njl8hU0ZEZEg823vjnYlfIj71Z+xKXI/K6gptrKamGruPbkTK2QRMGDqrTi8qUVvCBgYRUQNMTMzQ\nzsENOxPX43pBNsJ6j9Z5tsNEaQo7az7kSdRWKBVKRPYbi0Dvgdi0bznOZKfoxG/czsG/tnyIkIBh\niBk0DdYWtjJlSiQfNjCIiBpgbmqBUSGTAAA3i67jfE46CovztfEreecwIXKWXOkRkUyc7FzxesxH\nSD13CFviV6CkXHcissT035B+MQnPRbyMvj5hfAic2hQ2MIiIGqmdfQe0s++gs+xSbgaSMvc/cls3\nJ0+4t+vcTJkRkRwEQUBfn0Hw6xSIbQlrcCR9r068pFyNNbuX4GhGHCYMmQknlatMmRK1LDYwiIie\nwMgBE7D72A8QRbHB9W7ezmUDg6iVsrKwwaRhb6B/98HYELsM+bev6cQzs1OxaN1biBowEUP6jIFS\nyT+/qHVTSL3Db775Bl26dIGlpSWCgoKQkJAg9SGIiAyGo50LiktvwdbKXu+XhZklxg+ZiacHjJc7\n1VaFtYYMkZd7AN598StEhUyq04ioqq7E9kNr8cWGd5Cdd1amDIlahqRN6I0bN2Lu3LlYtmwZBg0a\nhKVLlyIqKgpnzpyBh4eHlIciIjIYM8d+XG/s0Kk92Ju8WWdZVXUFYgZNb+asWi/WGjJkpiamiBow\nAX19BmHjvmU4f+20Tjy34DKWbHwPYb2j4GYRADMTc5kyJWo+kvZgLFmyBDNmzMDLL78MX19ffP31\n1+jQoQOWLVsm5WGIiIzGUz2fxqiQSRgVMkk7moxCUMqclXFjrSFj4Orgjree+wQvDnsLVg+NJCVC\nxIGTO7E99d+4UpgpU4ZEzUeyHozKykqkpKTg3Xff1Vk+YsQIHD58WKrDEBEZhbNX03D49F7YWqkA\nABZmVgBEjB44Wd7EjNxj1RoDHr0nSO4EmoC5Np0AIOT3r/oUOP2InX/ch4HzPoeDrXMLZUbUvCRr\nYBQUFKCmpgaurrojJLi4uCAvL0/vNrUz6BorY88f4HswFHwP8pM6/9K7RVBUWaBMXYGCkhx0cw2E\nm33XZvmcvL29Jd+noXqcWkNkyJwL76D/1xvxqWMpAjsNgW+HflAIkj8iKxljOtcbS67GkGdT64zh\n/gYTERkxGwt7+LsNgIttR7jYeeDklQNyp0REBsq58A6qaiqRdGkPdqetxq2y/EdvRGTAJOvBcHZ2\nhlKpRH6+7n+K/Px8dOjQQe82QUGG0onZNLUtTWPNH+B7MBR8D/Jr7vzX/XoIXTr7IWLA0/Bw6dos\nx1Cr1Y9eqZV4nFpDZEwKSnOx8+R/MKRvDKIGTISZqWE8BG5M53pjydVY8gSaXmck68EwMzNDv379\n8Ouvv+os37t3L0JDQ6U6DBGRUYkZNB0nzh/GqYtHEXt8q9zpGL3HqjWiaLBfyUlJSE5Kkj0P5irD\nVwM0ogaxx3/C4nV/REZ2qlT/fYhajKS3SL399ttYvXo1/vOf/yAjIwNz5sxBXl4eZs6cKeVhiIiM\nhq2VCm88uxCB3Qbi2o0LcqfTKrDWUGvU26vuo+CFxflYtnUh1uz6O4rLimTIiujxSDoPxvjx41FY\nWIhPP/0U169fR8+ePbFz506OS05Ebdr+E7+gQJ2H4cHPy51Kq8BaQ63Ry9F/xqmLx7ApbjmKSgt1\nYsfPHkRGdipiBk1DSMAwCAY8MhoRIHEDAwBmzZqFWbNmSb1bIiKDID7i1gYAyC3IRnzqdjjYtQMA\naDQajIt4pblTa1NYa6g16tm1P7w79sSOI9/hwIkdEHH/fHOnohTrY5fiWGY8Jg6dBVfHjjJmStQw\nyRsYREStWeaVE9h/4hfYWtnD0twalmZWetcbFjwOrg7uLZwdERk7CzNLjIt4BUG+EdgQuxQ5BZd1\n4hdy0vHZ93MxIuh5DAsaB1MTU3kSJWoAGxhERE3QwakTAruFokZTjas3zqO31wC961mZW7dwZkTU\nmni298Y7k/6O+NSfsStxPSqrK7Sxmppq7Dq6ASlnEzAhcha6uQfImClRXWxgEBE9QF16C2evpTW4\njlKphEKhQE1NDRLTY+vEyyvvoHe3gQj2i2iuNImoDVAqlIjsNxaB3Qbih7jlyMhO0Ynn376Grzf/\nBSEBwxAzaBqsLWxlypRIFxsYRNSmVVSX4+z147iluQQAuFGUi4AuQejY7tFzVnTp4FdvzNbKXrIc\niahtc1K5YmbMR0g5m4Af969ESbnunASJ6b8h/WISng1/Cf18w/kQOMmODQwiatOqqitwqywfzlVO\nAACVtSNyC7JxvfAKqmuqEeI/FC58loKIZCYIAvr5hqG7Zx9sP7QGh0/v1YmXlKuxds9XOJYZjwlD\nZsJJ5SpTpkRsYBBRG/Djgf/CwsxSbyz3Ri5UVs5QKuueDiurylFRdbe50yMiajQrCxtMjHwDwX6D\nsWHfMuTfuqYTz8xOxaJ1byFqwEQM6TNG77mNqLnxt46IWqXth76Fye+Ftb2jB0J7DNe7XnJyMgAg\nKCioxXIjInpSXu4BeHfSV/jt+I/4NWkTamqqtbGq6kpsP7QWyVkHMClyNjzb+8iYKbVFkjYw/u//\n/g/r169HamoqiouLcfnyZXTq1EnKQxBRG6XR1OBuZXm98a0Jq2Fv46R97WDjhLDeo1oiNWpBrDNE\n95mamCJqwAT09RmEjbHf4HxOuk48t+Aylmx8D2G9R2H0wMmwNNc/rDaR1CRtYJSXl2PkyJEYO3Ys\n5s2bJ+WuiaiNyC3Ihihq6iwvLL6Bk+ePwMPFS+92fp0C0ddnUHOnRzJjnSGqy9XBHW+N+xRHz+zD\n1oOrcKeiVBsTIeLAyR04eSERLwx+Fb28QmTMlNoKSRsYc+bMAXD/lgMioqbauG8ZXBzcYWZiDh+P\nXjqx6NDJcLBtJ1NmZAhYZ4j0EwQBIQGRCOjSDz8dXIXkzP06cXVpIVb+8hl6eQ3AuIhX4WDrLFOm\n1BbwGQwiMhinLh5De0cPaEQNbhbl4oUhr8mdEhGRUbG1ssfUp+ch2G8wfoj7NwrV+TrxtAtHkXU1\nDdEDJyOsVxQUCqVMmVJrJoiiKEq90+TkZPTv31/vvbFq9f2xm8+dOyf1oYnIwN0suYac2xf0xkrK\nbyG464jfXwmwMOX9wo3h7e2t/V6lUsmYSctpqM4ArDVk+IKCg3VeJyclSX6M6poqpF09iPScIxBR\n9889Zxs3hHQbDUdrDmlLDWtqnXlkD8aHH36IRYsWNbhOfHw8wsPDG5EeEbV2GblJqKi+U2+85O5t\nBHUeDksz6xbMigwZ6wxR8zBRmqJv56Ho0i4AR87vREFpjk68oDQXO06shL97CHp7hMNEaSpTptTa\nPLIHo7CwEIWFhQ3uxMPDA5aW98eYb2wPhrFeaWsNw1ryPRiG1vAe4hL2Iv3aEXTtfO/qhkajQXTo\nZJmzarzW8DMw9vOq1HUGMJ7PxJh+/5irxB6ebVv6G0p0aDQ1SDi1Bz8f/hYVekbkc7JzxfihM9Hd\ns4/e7Y3iM/2dseRqLHkCTT+nPrIHw8nJCU5OTo9ajYiMnCiKervQ9dkQ+412SNic/Gvo5hqIkSFj\nmjM9asVYZ4ian0KhRHjvUejlNQBb4lfg5IVEnXhhcT6WbV2Ifj5heDb8ZdhZ28uUKbUGkj7knZeX\nh7y8PJw9exYAkJ6ejlu3bsHT0xMODg5SHoqIHkN5RZnO8IUPKiopRFzqNri36/rI/Tw4JCxH86GW\nxDpD9GTsbZzwcvSfceriMWyKW46iUt3ew+NnDyIjOxUxg6YhJGAYhId7WogaQdIGxr///W/89a9/\nBXBvuLTRo0dDEASsWrUKU6dOlfJQRPSAW8U3cKvk5iPXSzmbgA5OnWBmYqY3HjNoOtrZd5A6PSLJ\nsM4QSaNn1/7w7tgTO458hwMnduj0YN+pKMX62KU4lhmPiUNnwdWxo4yZkjGStIGxYMECLFiwQMpd\nElE90i4koqLqLgDg5PlEhDdi1urAbgPh5R4AJYclJCPFOkMkHQszS4yLeAVBvhHYELsUOQWXdeIX\nctLx2fdzMSLoeTgqOkOp4OwG1Dj8TSEyIkdO70XxnSIAQIE6DyOCnwcAdG7vy54HIiJ6LJ7tvfHO\npL8jPvVn7Epcj8rqCm2spqYau45ugMrSCSFeowAY/gPJJD82MIgMSN6tq0jKiIdSqf+/5t2KO4gZ\nNO3eC0FgTwQREUlCqVAist9YBHoPxA/7liMjO0Unri4vxJ7T30ItXkfMoGmwtrCVKVMyBmxgEBmA\nmppq/BC3HCZKUwzsMQwdG/GgNRERkdSc7FwxM+YjpJxNwI/7V6KkXK0TT0z/DekXk/Bs+Evo5xvO\nh8BJLzYwiJrJtoQ1MK3nYepaubm5AID8qrPw79wXvbsNbInUiIiI6iUIAvr5hqG7Zx9sP7QGh0/v\n1YmXlKuxds9XOJYZj/FDXoezqr1MmZKhYgOD6CF3K8uh0dQ8cr3krP24VXwDZqYWeuNOdq4Y1Gtk\nw/swokl2iIiobbGysMHEyDcQ7DcYq3csgbpcd0jbzOxULF73R0QNmIghfcbUe3svtT38TSD63d3K\nchSor+O35B/Rub1vo7aJGjAR5maWj16RiIjISHm5ByA68FWcvnYYp3MPo6amWhurqq7E9kNrkZx1\nAJMiZ8OzvY+MmZKhYAOD2qzcgmwUFudrX+fcvASFoEBYryh4uQfImBkREZFhUSpM0LtTOKKHTMDG\n2G9wPiddJ55bcBlLNr6HsN6jEB06BRa8+NamSdbAuH37Nj7++GP89ttvyM7OhrOzM6Kjo/Hpp5/C\n0dFRqsMQNcq5a6dwvfBKg+tkZKdiVMgk7Wt7Gye42LuxR4LIQLHOEMnP1cEdb437FIlnYrHt4Grc\nqSjVxkSIOHByB05eSMQLg19FL68QGTMlOUnWwMjNzUVubi6++OIL+Pv749q1a5g9ezYmTZqEPXv2\nSHUYokZJytyPIX1iYGulqnedfj5hsLa0a8GsiOhJsM4QGQZBEDAwYBh6dAnCTwdWITlrv05cXVqI\nlb98hl5eAzAu4lU42DrLlCnJRbIGRkBAALZs2aJ93bVrV3zxxReIjo5GaWkpbGxspDoUtVG3im9g\n/4lftD0MtSMw3ag+V2ddUaOBqYkpbNiAIGo1WGeIDIutlT2mjpyH4O6D8UPcv1GozteJp104iqyr\naYgeOBlhvaKg4NxNbUazPoOhVqthbm4OKyur5jwMtQKFxfn4LelH2Frb17tOZVUFenmFwMvdHwBH\nYCIi1hkiQ9Ddsw/en/w1dh/7AftStuqMxFhRWY4t+1ciOXM/Jka+Afd2neVLlFqMIIqi2Bw7Lioq\nQnBwMEaPHo1//OMf2uVq9f0JW86dq3vlmVo/jaips6y8shQZuUehVJjidlk+IvzGQangGAREjeHt\n7a39XqWq/7bA1qa+OgOw1pDhCwoO1nmdnJQkUybSul2WjyPnd6KgNKdOTIAAf/cQ9PYIh4nSVIbs\n6HE1tc4oHrXChx9+CIVC0eDXgQMHdLYpLS3FM888Aw8PD3z++eeP8TaoNamsvouSu7e1X/vObMTp\na4d0vi7cOAkzEwsoFUo427pBEB75q0lErQTrDFHr4WDtipG9pqF/15EwVepONitCRHrOEWxPXY7c\n2xdkypBawiN7MAoLC1FYWNjQKvDw8ICl5b374ktLSzFq1CgIgoBdu3bV6bZ+8KqSsV5paw235jzp\neyhU5+NWyc1GrZuUGY8u7X219166OLijS4fGzTPREP4cDIOxvwdjzx8w/vOq1HUGMJ7PxJh+/5ir\nxARB93Xz3FAimcf5TItKC7E5fgXSLiTqjffzDcdz4S/B1qr+26Mfh1H8/GE8eQJNP6c+8h4UJycn\nODk5NergJSUliIqKavCkT8ZLFEUkZcYDANIuJCK8d3SjtgvyjUC3jgFQsFeCiPRgnSFqnextnPBK\n9J+RduEoNsX/H9SluhcSjmcdQMblFMQMmoaQgGEQHm50kdGS7Cb3kpISjBgxAiUlJdi6dStKSkpQ\nUlIC4F7xMDXlvXaGqrS8GIdONWaIRxGl5cUI7z0aXTr4oZ19h2bPjYioFusMkXHq5TUAPh69sOPI\ndzhwYgdE3O+tuVNRivWxS5GUGY8JkbPh6uAuY6YkFckaGMePH8fRo0chCAJ8fO5PEy8IAuLi4hAe\nHi7VoUgCGbnHUFFdjhvV51BeUQYvN3/07Nr/kdsJgsBh5ohIFqwzRMbLwswS4yJeQZBvODbEfoOc\ngss68fM56fjsuzkYEfQ8hgWNg6kJLxgYM8kaGIMHD4ZGU3d0IJJf+qVkXMzNgFJ5/8ddUXUHgZ6D\njeK+PyIigHWGqDXwbO+DdyZ+ifgTP2Nn4npUVVdqYzU11dh1dANSziZgQuQsdHMPkDFTehIcB7SV\nu5CTjuNZB/H84FdhZXF/EqraB4uIiIiIWpJSaYLIfs8isFsofohbjozsFJ14/u1r+HrzXxASMAwx\ng6bB2sJWpkzpcbGBYaQqqypQVXOv1b9x3zK0d/TQu55Go0F06GSdxgURERGR3JxUrpgZ8xFSzibg\nx/0rUVKu1oknpv+G9ItJeC7iZfT1CeND4EaEDQwjUVVdibxb17SvE9J2wc3ZEwAQ4j8M/p37ypUa\nERER0WMRBAH9fMPg5xmI7QlrcSR9r068pFyNNbuX4GhGHCYMmQknlatMmVJTsIFhgM5cTkF1TZXO\nMnXZLdwsuq69H7FH1+BGPZRNREREZOisLWwxadgb6N99MDbELkP+7Ws68czsVCxa9xZGhUzC4MBn\ndJ4rJcPDn44BOXJ6LyqrK3DtxkWEB+rOMeFg2w59fQbxPkQiIiJqtbzcA/Dui1/ht+M/4tekTaip\nqdbGqqorsS1hDZIz92Ni5Gx4tvdpYE8kJzYwZHKz6DqOZeyD8MDkc3cr7uDp/i8gxD8S5maWMmZH\nREREJA9TE1NEDZiAvj6DsDH2G5zPSdeJ5xRcxpKN7yGs9yiMHjgZluaccNPQsIHRwsrulmD30Y2o\nqanGAP+hbH0TERER6eHq4I63xn2KxDOx2HZwNe5UlGpjIkQcOLkDJy8k4oXBr6KXV4iMmdLDJG1g\nvPrqq4iLi0Nubi5sbGwQGhqKxYsXo3v37lIexuD9fOjbeu8NrK6pRnfPPvDv3K+FsyIiMn6sM0Rt\niyAIGBgwDD26BOGnA6uQnLVfJ64uLcTKXz5DL68BGBfxKhxsnWXKlB4kaQMjODgY06dPh4eHBwoL\nC7FgwQIMGzYM2dnZMDFpvZ0lNZoaAMCuxPVQKJRQ2TghvPcombMiImp92mqdIWrrbK3sMXXkPAR3\nH4wf4v6NQnW+TjztwlFkXU3DM6FTYCG2g+KBW9Cp5Ul6Nn7ttde033fq1AmffPIJAgMDcenSJXh7\ne0t5KFmVV5ahWlOJAnUeAGBj7DJ06xgAWyt7RDz0cDYREUmnrdQZItKvu2cfvD/5a+w+9gP2Hf8J\nGlGjjVVUlmNz/Ao427hjYDde6JVTs13uKSsrw6pVq+Dt7Y0uXbo012GaXVV1JS7nZeksS768F272\nXXEx997H98xTf0An125ypEdE1Ga1ljpDRE1jZmqOMU/9Af18wrBh3zfIzjurEy8ozcEvJ1aiXFmI\nkQMmwMzUXKZM2y5BFEVRyh1+8803eO+991BWVgYvLy/s2rUL3brd/+Nbrb4/S+O5c+ekPHSzuFNR\njDO5R9HR8f6VMaXCBO1sO8qYFRHRfQ9euVepVDJm0jIeVWcA46s11PYEBQfrvE5OSpIpE+OmETU4\nm5eC1Ox9qKqprBO3MbdHiFcU3By8ZMiu9WhqnXnkDWoffvghFApFg18HDhzQrj9lyhScOHEC+/fv\nh7+/P6KiolBSUvKYb6flpV09qPOVeT0ZNhb2aK/qrP1i44KISDptrc4QkXQUggJ+HYIQ02cmOjn6\n1omXVhThtzPrcTDrJ5RXlsmQYdv0yB6MwsJCFBYWNrgTDw8PWFrWnbehqqoKDg4OWLp0KaZNmwZA\n96qSIV1pSz13CNcLr8Dc1AKD+4zRiQmCoPOwUHJyMgAgKCioRXOUEt+DYeB7kJ+x5w8Y7nm1saSu\nM4DxfCbG9PvHXCUmCLqvpb2hRHJG8ZkCSLuQiO9/XYo7lXUvOliZ2yAmbDpC/CMhPPz5y8BYPlOg\n6efURz6D4eTkBCcnp8dKRqPRQBRFaDSaR6/cwk5dPIbsvLNQKJQAgKLSQrw47E2ZsyIiantaa50h\nopbXyysEZX1qkHolHlnXkyHifsPtTkUp1v/2v0jKiMOEyNlwdXCXMdPWTbKHvC9cuIDNmzdj+PDh\ncHZ2xrVr1/DZZ5/BwsIC0dGGN7KSUqGEytoRYRxOlojIKBhbnSEieZiamKN/16cxOnw8NsQuRU7B\nZZ34+Zx0fPbdHIwIeh7DgsbB1MRUnkRbMckaGObm5ti/fz+WLFmCoqIiuLq6IiIiAkeOHEG7du2k\nOkyT3blbir3JW2BqYqYbEAEvd395kiIioiYz1DpDRIbJs7033pn0d8Sn/oydid+jqvr+Q+A1NdXY\ndXQDUs4mYELkLHRzD5Ax09ZHsgZGx44dsXPnTql298TUpbdQfKcIe45thG+nQIT1ipI7JSIiegKG\nVhn2I28AACAASURBVGeIyPApFUpE9huLwG4D8UPccmRkp+jE829fw9eb/4KQgGGIGTQN1ha2MmXa\nurTKaQ4vXc/CTwf/i9slNzEi+AU2LoiIiIjaMCeVK2bGfIRpI9+GrWXdh5QT03/DorVv4njWAUg8\ng0Ob1GwT7bWUzOwTyL99TWfZ+Zx0PN3/BXRs11WmrIiIiIjIkAiCgH6+4fDz7IPtCWtxJH2vTryk\nXI01u5fgaEYcJgyZCSeVq0yZGj/jb2BcSUVlVQWe6jkS9rb3RiHp330ILM2tZc6MiIiIiAyNtYUt\nJg17A8HdB2Nj7LI6F6ozs1OxaN1bGBUyCYMDn4FSafR/Lrc4o75FKiFtN8xMLaARa6BQKGBtYQtr\nC1s2LoiIiIioQd3cA/Dui18hKmRSnUZEVXUltiWswZcb3kF23lmZMjReRtkki0vZjvLKMlRVVyBm\n0HS50yEiIiIiI2RqYoqoARPQ12cQNsZ+g/M56TrxnILLWLLxPYT1HoXRAyfD0txKpkyNi1H1YIii\niJqaamReOYFRIZPYuCAiIiKiJ+bq4I63xn2KF4e9BStzG52YCBEHTu7AonVvIe1CokwZGhejaWCU\nlRfj2s1LWPHzIoRzcjwiIiIikpAgCAgJiMRfpv4vgvwi6sTVpYVY+ctnWPnLZygqLZQhQ+MheQND\nFEVERUVBoVBgy5Ytku03/sTPuF6YjbHhMxDQJUiy/RIRkXFprjpDRAQAtlb2mPr0PMwaOx9OdnVH\nkkq7kIj/9+2bOHByBzSaGhkyNHySNzD+/ve/Q6lUArjXEnxShcX5OHpmH/IKr6J/9yFo7+jxxPsk\nIiLjJXWdISLSp7tnH7w/5WsM6/ccFILun8wVleXYHL8CX216Hzk3L8uToAGTtIGRlJSEr7/+GqtW\nrZJkf8cy4rA9YS3srB0wNmyGJPskIiLjJXWdISJqiJmpOcYMmor/mfR3eLp614ln553FFxv+hO0J\na1FZVSFDhoZJslGkSkpK8OKLL2LFihVo167dE+2rUJ2PQ6f2oKLqLqaNfBuCoOBVKiKiNk7KOkNE\n1BTu7bpg3vjPkHBqD34+/C0qKsu1MY2mBr8d/xGp5w5h/NCZ6O7ZR8ZMDYMgSjQf+uTJk+Hs7Ix/\n/vOfAACFQoHNmzfjueee01lPrVZLcTgiItJDpVLJnUKzaWydAVhriIiaS2PqTIM9GB9++CEWLVrU\n4A7i4uJw5coVpKWlITk5GcC9B/Ae/JeIiEgf1hkiotanwR6MwsJCFBY2PAyXh4cHZs+ejbVr10Kh\nuP9IR03Nvdm1Q0NDceDAAe1yXlUiImo+xtaD0Rx1BmCtISJqLo2pM5LcIpWbm4uioiLta1EU0bNn\nT3z11VeIiYlB586dn/QQRETUhrHOEBEZD0ke8nZzc4Obm1ud5R4eHjzpExHRE2OdISIyHkYzkzcR\nERERERk+yUaRIiIiIiIiYg8GERG1eqIoIioqCgqFAlu2bJE7Hb1effVVdOvWDVZWVnBxccHYsWOR\nkZEhd1p13L59G2+99Ra6d+8OKysrdOrU6f+3d+fRUdZ5vsc/VZV9qywkYUkICAEF2ZoQJbQsCig0\norSDW7tPy3hdGpser9ptj+MfHrt19Db3DJy+MtOIICrKNNoKgktIREAIEBAQEpaEhBCWrJUQyFLP\n/UNNm05CAlTyq6q8X+fkUHmoKt4pQqhv/Z7nKT366KMqLy83ndam119/XVOmTFF0dLTsdruOHTtm\nOqnZ4sWLNXDgQIWGhiotLU2bNm0yndRKdna2Zs+eraSkJNntdi1btsx0UrteeukljRs3Tk6nUwkJ\nCZo9e7b27dtnOquVRYsWadSoUXI6nXI6ncrIyNDatWtNZ3XopZdekt1u1xNPPNHhdRkwAAB+79VX\nX5XD4ZAkr33j1nHjxmnZsmU6cOCA1q9fL8uyNHXqVDU2NppOa6GkpEQlJSV65ZVXtHfvXq1YsULZ\n2dm66667TKe1qa6uTjfddJNeeOEF0yktvPvuu3ryySf13HPPKTc3VxkZGZoxY4aKiopMp7VQW1ur\nkSNHauHChQoNDfXafz+SlJWVpccff1xbtmzRF198oYCAAE2dOlUVFRWm01pITk7Wyy+/rF27dmnH\njh26/vrrdeutt2r37t2m09q1detWLVmyRCNHjuzc94AFAIAf27Ztm5WcnGydOnXKstls1urVq00n\ndcru3bstm81m5eXlmU7p0Nq1ay273W65XC7TKe3avn27ZbPZrMLCQtMplmVZVnp6ujVv3rwW21JT\nU61nn33WUFHHIiIirGXLlpnO6LSamhrL4XBYH330kemUDsXGxlqvv/666Yw2VVZWWoMGDbI2btxo\nTZ482XriiSc6vA0rGAAAv+VyuXT33XdryZIlio+PN53TabW1tVq6dKlSU1M1cOBA0zkdqqqqUnBw\nsMLCwkyn+IT6+nrt3LlT06dPb7F9+vTp2rx5s6Eq/1NdXS23262YmBjTKe1qamrSO++8o3Pnzmni\nxImmc9o0b948zZ07V5MmTer0m5t65DS1AAB4o0ceeUQzZ87UjTfeaDqlUxYvXqynn35atbW1GjRo\nkNatW6eAAO/+r7qyslK///3vNW/evBZvhIj2nTlzRk1NTUpMTGyxPSEhQaWlpYaq/M/8+fM1ZswY\njR8/3nRKK998843Gjx+v8+fPKzQ0VKtWrdLQoUNNZ7WyZMkSHTlyRCtXrpTU+V1M+UkAAPApzz33\nnOx2+wU/srKytHz5cu3Zs0cvv/yyJDW/8tbZV+C6q/XH70J+zz33KDc3V1lZWRo2bJhmzJghl8vl\nla2SVFNTo5tvvrl5n/Lucimt6FkWLFigzZs3a/Xq1V553MiVV16pPXv2aNu2bXr88cd15513Kicn\nx3RWCwcPHtTvfvc7vfXWW83HsFmW1amfoZymFgDgU8rKylRWVnbB6yQnJ+vRRx/Vm2++2eJV9aam\nJtntdmVkZHTLE9DOtoaGhrba3tDQoJiYGC1atEj3339/VyU2u9jWmpoazZw5UzabTevWrevW3aMu\n5XHNyclRenq6CgoK1L9//65OvKD6+nqFh4frnXfe0W233da8/bHHHtP+/fuVmZlpsK59kZGRWrRo\nke677z7TKRf061//WqtWrVJmZqaGDBliOqdTpk2bpqSkJC1dutR0SrM33nhDDz30UPNwIX33M9Rm\ns8nhcKi2tlaBgYFt3ta7110BAPgHcXFxiouL6/B6L774op566qnmzy3L0ogRI/Tqq6/qlltu6crE\nZp1tbYvb7ZZlWXK73R6uatvFtLpcLs2YMcPIcCFd3uPqDYKCgjR27Fht2LChxYDx6aefau7cuQbL\nfN/8+fP13nvv+dRwIX33xL27/q131pw5c5Sent78uWVZevDBBzVkyBD99re/bXe4kBgwAAB+qm/f\nvurbt2+r7cnJyRowYED3B13A4cOH9f7772vatGnq1auXiouL9Yc//EEhISGaNWuW6bwWXC6Xpk+f\nLpfLpTVr1sjlcjXvxhUXF3fBJx0mlJaWqrS0VHl5eZKkffv2qby8XCkpKUYP/l2wYIHuvfdepaen\nKyMjQ3/+859VWlqqRx55xFhTW2pra5Wfny/pu6G3sLBQubm5iouLU3JysuG6lh577DGtWLFCa9as\nkdPpbD6eJTIyUuHh4Ybr/u6ZZ57RrFmzlJSUJJfLpZUrVyorK0uffPKJ6bQWfnifjh8LCwtTTEyM\nhg0bduEbd9k5rQAA8DLeepraoqIia8aMGVZCQoIVFBRkJScnW/fcc4918OBB02mtZGZmWjabzbLb\n7ZbNZmv+sNvtVlZWlum8Vp5//vkWjT/86g2nW128eLE1YMAAKzg42EpLS7O+/PJL00mt/PD3/Y9/\n5w8++KDptFba+r602WzWCy+8YDqthQceeMBKSUmxgoODrYSEBGvatGnWhg0bTGd1SmdPU8sxGAAA\nAAA8hrNIAQAAAPAYBgwAAAAAHsOAAQAAAMBjGDAAAAAAeAwDBgAAAACPYcAAAAAA4DEMGAAAAAA8\nhgEDAAAAgMcwYAAAAADwGAYMAAAAAB7DgAEAAADAYxgwAAAAAHgMAwYAAAAAj2HAAAAAAOAxDBgA\nAAAAPIYBAwAAAIDHMGAAAAAA8BgGDAAAAAAew4ABAAAAwGMYMAAAAAB4DAMGAAAAAI9hwAAAAADg\nMQwYAAAAADyGAQMAAACAxzBgAAAAAPAYBgwAAAAAHsOAAQAAAMBjGDAAAAAAeAwDBgAAAACPYcAA\nAAAA4DEMGAAAAAA8hgEDAAAAgMcwYAAAAADwGAYMAAAAAB7DgAEAAADAYxgwAAAA2lFQUCC73a4H\nH3zQdArgMxgwAAAAOmCz2UwndOiHYWjKlCmmU9DDBZgOAAAA8FZJSUk6cOCAnE6n6ZQO/TAE+cIw\nBP/GgAEAANCOgIAADRkyxHRGp1iWZToBkMQuUgAAAO1q6xiMBx54QHa7XVlZWXr//feVnp6u8PBw\nxcXF6a677lJJSUmr+5k8ebLsdruOHj2q//iP/9DQoUMVGhqq/v3761//9V9VU1PT6jYX2t3p3//9\n32W325WdnS1JeuONN3TFFVdIkjZu3Ci73d788cILL3jioQA6jRUMAACADrS129HixYv14Ycf6pZb\nbtGUKVO0detWvfvuu9q9e7dyc3MVFBTU6jbz58/XV199pTvuuENOp1Nr167Va6+9pk2bNik7O7vV\nbTq7u9OYMWM0f/58LVy4UAMGDNADDzzQ/HuTJ0++qK8VuFwMGAAAAJdg/fr1ysnJ0fDhw5u3/eIX\nv9Dbb7+tDz74QHPnzm11m61bt2r37t1KSkqSJL344ou67bbb9MEHH+i1117TM888c0kto0aN0pNP\nPtk8YPzbv/3bpX1RgAewixQAAMAl+NWvftViuJCkhx9+WJK0ffv2Nm8zf/785uFC+m43qD/+8Y+y\n2Wz6y1/+clk9HIMBb8GAAQAAcAnS0tJabftheKioqGjzNpMmTWq1bciQIUpISNDhw4dVW1vr2UjA\nAAYMAACASxAdHd1qW0DAd3ufNzU1tXmbxMTEC26vrq72UB1gDgMGAABANzl58uQFt0dFRbXY3tjY\n2Ob1KysrPRsGeBADBgAAQDfZuHFjq20HDx7UyZMnNXjwYIWHhzdvj4mJUVFRUZv309YxHg6HQ1L7\nqydAd2HAAAAA6CYLFy5sMTQ0NTXp6aeflqQW77UhSddee60KCwu1bt26FtuXLFmiLVu2tDqFbUxM\njCS1O5QA3YXT1AIAAHSTCRMmaPTo0br99tsVFRWldevWae/evUpPT9dvfvObFtd96qmntH79es2Z\nM0e333674uPjtWPHDu3YsUOzZs3SRx991OL6ERERysjI0ObNmzV79myNGTNGgYGBmjRpkq677rru\n/DLRw7GCAQAAcBFsNlun3wDvH2/3pz/9Sc8++6wyMzO1cOFCVVZWasGCBfr8888VGBjY4vqTJ0/W\nhx9+qNGjR+v999/X0qVLFR0dra+//lpjx45ts2H58uW69dZbtWXLFr344ot6/vnnlZmZeclfK3Ap\nbBYnTQYAAOhSkydPVnZ2tgoKCtS/f3/TOUCXYgUDAACgG1zKqgfgixgwAAAAugE7jaCn4CBvAIDf\nqaqqMp0AtNDU1CSbzabq6mq+P+HTnE5nh9fhGAwAgN/hCRwAdI3ODBjsIgUAAADAY9hFCgDg1zrz\napspOTk5kqS0tDTDJR2j1fN8pVOitSv4Sqd08avCrGAAAAAA8BgGDAAAAAAew4ABAAAAwGMYMAAA\nAAB4DAMGAAAAAI9hwAAAAOhirrNVyi/+RqcqSkynAF2O09QCAAB0scPH96mu/qw+z/mrhvYfrSk/\nmS1JcltuNTU1ym53yGazqaA0T8dOHlJCdF8lxvZTTGS84XLg4jFgAAAAeNjGXX9TZc0ZXdH3KvWO\nTdaZqlING/ATjUmdoA3b35frbJW2HVmvqroyfXvmS6UPu15nKk/o28JdmjPxIR0p2a/Nezeol7O3\nrhs1UzGRvUx/SUCnMWAAAAB42NnzNbpu5EwdOr5P2bvXamj/UYqJTFBIUKgcdrsyd32o2PBEpV9x\nowYOSVZe0TeKjUrQHTf8LznDYxUf3UfXDLtBuw9tVVn1SYWFRCg4MERud5Nq6qrlsDsUHhpl+ssE\n2sSAAQAA0IXCQiI04or05s9vTL9dbrdbubm7JUlxUYkaPzyxzdv27ZWi/QU7tO/odt3y0wdUVVuu\nv21eIYfNoV9M/1W39AMXi4O8AQAAukh9wzm53U0ttgU4AhUUGCy7reOnYfHRfTRp9CwFBgQ3b0vt\nd7Viojg2A96LFQwAAAAPyC/eq31HcxQUGKzy6lMKC4lQo7tRfXsN8Mj9V9aU6fjpAo/cF9CVGDAA\nAAA8oKauStcMu1594vo3b5syZrZH7ruypkx/+2q5xqRO0ND+o7V1/2ceuV+gKzBgAAAAXKaPNq/Q\n+YZzSkkc0iX3P374NLndjRrUb3iX3D/gSQwYAAAAlyBz54eqrDmjuvqzCgkK022Tftllf9bAPkPb\n/b3S8iK53W717ZXSZX8+cDEYMAAAAC5BXX2t5kx8yHSGNmx/X41NDXpo5v82nQJI4ixSAAAAF239\ntlUqKM0z9udHhcVoxYaFKjlToF7O3uodm2ysBfhHNsuyLNMRAAB4UlVVVfPl/Px8gyXwV7nHsjS6\n/ySjDcfKDupsfbXOnnfJbncY74H/Sk1Nbb7sdDo7vD4rGAAAAD4oMaq/4iL6KLX3GNMpQAscgwEA\n8GtpaWmmE9qVk5Mjybsbf0BrS6ca8y/7/j3ZWbf1TJd+vfz9e56vdEotV4U7gwEDAACgk05VHNf+\ngp1qbGwwndKSJTU2Nchus8tud5iuQQ/HgAEAANBJxaePKqV3qvr1Gmg6pYUh/Ufq4y0rFR/dRxlX\nTzedgx6OYzAAAAAu4Oy5Gn26fbXe2vB/tb9gh8JDohQUGGw6q4XB/YZratrPdaLsmFZn/ZfpHPRw\nrGAAAAC0Y+3Wt2VZloYNGKvJY2YrMCDQdFK7wkMiddukX+qDTct0srxYsVEJCgwIMp2FHogVDAAA\ngAv42fi7NbDPUK8eLn5s2ICx2rLvU5WWF5lOQQ/FCgYAAIAfSU26Wufqz5rOQA/GCgYAAEAblq59\nRQ2N501nXLKGxno1eNvZrtAjMGAAAAC0ITE2Sbf89AHTGZekd2yyjp44oI+3rDCdgh6IAQMAAOBH\nvi3cpb9m/0VxUQmmUy5ZfHQf3TB2joICQ+Q6W6WT5cU6e77GdBZ6CI7BAAAA+N7GXX9T8ekjujH9\ndsVH9zGdc9lq6qr18ZYV6tdroOx2hyaMuNF0EnoAVjAAAAC+d/Z8je6ZPt8vhgtJGnFFuoYPHKex\nQyfqZMVxrfriz6aT0AMwYAAAAPipq1LGaMQV6QoLidDPJz6kiDCn6ST0AAwYAAAAPYRluVV3vlZu\nd5PpFPgxBgwAANDjVbhO6+3PFslhd5hO6VLJCYP01+y/qPj0UdMp8GMc5A0AAHq0E2VFOlKyX4OT\nhmvclZNN53SpkYOulc3G68voWgwYAACgR9uZl60RV1zj06elBbwJIywAAOjRbDa7+icOVnholOmU\nbuGwB2jrvs+0M2+T6RT4KZtlWZbpCAAAPKmqqqr5cn5+vsES+ILcY1ka3X+S6Yxul3P0U13ZZ5zC\ng52y2Wymc+DFUlNTmy87nR2fiYwVDAAAgB4oLqKvth/dYDoDfogVDACA3/nxCkZnXm0zJScnR5KU\nlpZmuKRj/tZ6vr5Orroqrd/2ngLsAbrjhv/VXXnNvOEx/XzHX3Wu/qyu7D9ag/oNb/d63tDaWb7S\n6iud0sX/TOUgbwAA0OPsPZqj8upT+umIm5TSO7XjG/ipG8bOUVVtufYe2X7BAQO4GAwYAACgRxo5\n6BolxiaZzgD8DsdgAAAAAPAYBgwAAIAerrS8SIWleaYz4CcYMAAAAHqw8JBIjR48XtsPbDSdAj/B\ngAEAAHoEt+XW+fo6fbjpTR098a2CAoNNJ3mFAEegBvUbrrCQSNMp8BMc5A0AAHqEM5UntG7rO7r6\ninSNHXqd6Ryv09B4XsWnjyghuh/DFy4LKxgAAMDv5Rfv1dZ9n2v4wDSGi3aMHDReuflbdKLsmOkU\n+DhWMAAAgN/LL/pG09PnKtARaDrFaw3sM1Rnz7lMZ8APMGAAAAC/dqTkgCpryxQSFGo6BegR2EUK\nAAD4tS17N2jclZNMZwA9BgMGAADwazFR8UpNGmE6wyf0cvbW3qPb9cGmN0ynwIcxYAAAAECSlBib\npJ+Nv1tnqk7qw6+Wy225TSfBBzFgAAAAoIV//tnTCgwI0u5DW3S68oTpHPgYDvIGAAB+Kb/4G+0v\n2KmYyF6mU3zShKtv1MmKIh04lqtQxZvOgQ9hwAAAAH6pqqZc44dPVUJMP9MpPikqPFqSVFpebLgE\nvoZdpAAAgN/ZdmS9ik8fVUhQmOkUn1fpOqOqs2WmM+BDGDAAAIDfcFtuNTTVK9ARrFuve0BR4TGm\nk3xaaHCYUnqnamfhF9p3fIuqastNJ8EH2CzLskxHAADgSVVVVc2X8/PzDZagu1WdLdPuomwlxw7R\nwPjhpnP8RmNTg45XHFJ4cJR6RbLLWU+TmprafNnpdHZ4fVYwAACAX0mOTWW48LAAR6Ac9kDTGfAR\nHOQNAPBraWlpphPalZOTI8m7G3/gK60nK44re+t3p1X19lZfeUylv7deddVVSuk9xHDNhfnK4+or\nnVLLVeHOYAUDAAAAgMewggEAAPzCFzvXqKbOpfDgjvcRx8Wz2Wzasu9TVbjOaHRqhukceDFWMAAA\ngF84V1+n2RPuVUJUsukUv9QvZpDuvOExlZQVmk6Bl2PAAAAAAOAxDBgAAADotPLqU9p9aKt4pwO0\nhwEDAAAAnXbD2J/r4LFcWZbbdAq8FAMGAAAAOq1PXDLvkI4L4ixSAADAZx07eUjfHPlaDY31Kqs6\naToHgBgwAACADztZcVzjrpyiwIAg0ykAvseAAQAAfErtOZe27c/UkZL9igiL1uB+wxUT2ct0Vo9y\nRd+r9OFXy1V3vlajUzN0VcoY00nwIgwYAADAp9ScrVJIcJjumT5fwUGhpnN6pCHJIzWo33CdrizR\nlr2fMmCgBQ7yBgAAPuPs+RpVn61UUEAww4VhDrtDvWOT1djUqI+3vKXy6tOmk+AlGDAAAIDP+Dzn\nrzpdWaL+iYNNp+B7c6fM08A+V6r6bIXpFHgJdpECAAA+w+EIUMbV001nALgAVjAAAAAAeAwrGAAA\nwCcs++Q1RYY6TWegDaHBEdqZt0n7C3aotKxI98/4jRx2h+ksGMKAAQAAfEJ8dB/NvPYu0xlow8A+\nQzWwz1BJ0vptqwzXwDSbZVmW6QgAADypqqqq+XJ+fr7BEnhCRe1JHSs7KLfVpDEpU0znoAN7ir7U\n1UkTZLexJ76/SE1Nbb7sdHa8isgKBgAA8GoVtac0MH64IkNiTacA6AQGDACAX0tLSzOd0K6cnBxJ\n3t34A5Ot1oEaDeg9VPHRfTp1fV95XH2lU7q41jL3EYXEWuoXn6y4qMSuTmvFVx5XX+mUWq4KdwZr\nVwAAAPCYa4dNVVBAsPYd3WE6BYawggEAAACPcUbEym536FRliekUGMIKBgAA8FofbnpThaV5CgoI\nNp2CS5C9e63WfLlUpyoYNnoSVjAAAIDXCggI1Oyf3mc6A5fg28KdCgkM1birpqj2nMt0DroRAwYA\nAAA8KjLMqX+Z/ZwkaX/BTsM16G7sIgUAAIAuVVNXxSpGD8IKBgAA8Cp5RXuUV/SNzp5zKTAgyHQO\nLlOfuGR9W5irDdve05yJD5nOQTdgBQMAAHiVQ8X7NG3cbbpt8sM8IfUDMZHxyrh6mmrqqvU/Wf+t\nJneT6SR0MQYMAADgNf7ro5fktpoUHBgih91hOgcedO+NTyoiNEqW5Tadgi7GLlIAAMBr9O01QDOv\nvct0BoDLwIABAACMa2hsUH3jOckyXQLgcjFgAAAAo06UFWlX3iZZsnRVyhjTOQAuEwMGAAAwatOe\ndRo24CdKTRqhoEDesdufDex7lT7e8pZiIhM0cdRM0znoIhzkDQAAjAoPjdTwgWkMFz1AatLVuin9\nDlXWlGn5+j+ZzkEXYcAAAABAtwkOCtXsCfcqzploOgVdhAEDAAAAgMcwYAAAAADwGA7yBgAA3a74\n9BEVlubr+JkCRYRGmc6BIX/N/osyrp6uxNgk0ynwIJtlWZxxGgDgV6qqqpov5+fnGyxBe3YVZmpA\nr+EKD45SUECI6RwYUlJxWOca69Q7KkVhwZGmc9CO1NTU5stOp7PD67OLFAAA6Faf7lupAHugYsIT\nGC56uJjwRNlk0/6Sr02nwIPYRQoA4NfS0tJMJ7QrJydHknc3/sCTraca8zXz2rsu+37a4yuPq690\nSl3fuirz/6lXUpT6Jw6W3XZ5r3/7yuPqK51Sy1XhzmDAAAAA3aKqtlwny4/LstymU+BlxqRm6FDx\nXu05tFW945KVftUU00m4DAwYAACgWxwozFWAI1DXDLvBdAq8TGrSCKUmjVDtOZdyDmSZzsFl4hgM\nAADQ5VZ+9p86VXFcV/YfpV7O3qZzAHQhBgwAANDloiPidPOEexXOKWlxAXabXQWleVq79W3TKbgM\nDBgAAKDLVNWW629fLVdNXbXpFPiA0OBw3X/TAtWdr9XHW95SWdVJ00m4BByDAQAAusSew1/rwLFc\nDU0epVGDrzWdAx9y26Rf6kBhrsqqTyk6Ik4OB09ZfQkrGAAAoEsUnz6i26f8C8MFLkl8dB8VlB7U\nmk1vmE7BRWLAAAAAHvfhV8t14kyh6Qz4sDhnoqaP+yeFBoebTsFFYsAAAAAeF+AI0D/PesZ0BvxA\nefUp/U/2X9TU1Gg6BZ3EgAEAADym7nytTpYXS5bpEviLe6bPV0piqj7e+rZ25m0ynYNO4IgZ7fK+\nYgAADcBJREFUAADgEXsOb9Wew18rOWEQx13Ao8YOvU5XpYzRtm8zTaegExgwAACARxScyNPNGfcq\nIswph91hOgd+JsARqOqzlfrvj/+omdfepdioBAUHhpjOQhvYRQoAAHhEQECgnBGxDBfoEkGBwZo9\n4V5d/5NbtDPvSxWfOmI6Ce1gBQMAAFyWM1Wl+mLHGjkjYk2noAcY2OdK1TecN52BC2DAAAAAl2zT\nnk90urJEY4ZMUGrSCNM5ALwAu0gBAIBLsmH7+9p3NEdzJj7EcIFuFRPZS98W7tLbny1SyZlCuS23\n6ST8CAMGAAC4KAWlefp4y0o1uZv0S97rAgYkxPTTrIxfaNTga/VZzv+osDRPrrNVprPwPZtlWZyp\nGgDgV6qq/v5EIz8/32CJfzp0Mle9nQMUERJtOgVQdV25zriO61zjWQ3re43pHL+UmprafNnpdHZ4\nfVYwAAAA4LOiQmOVFDtEZ1zHtftYtukciBUMAIAf+vEKRmdebTMlJydHkpSWlma45MLO19dp+47t\nCnQE6XjdPlmyNH3cXEWGeedj6yuPq690Sr7Tunbr20oI+O7Vdm9v9ZXHVLr4n6mcRQoAAFzQJ9tW\nqex0heob6zRq2DiNHz7VdBLQppCgUOUWZEmSKnRM09J+brioZ2LAAAAAFxQYEKSRyT+VJKUN9/5X\nW9FzXf+TWxXl/m5l4FQjx1+ZwjEYAACgldLyIn31zXpt3fe5TlWUmM4BLtrZczV65/NFqq6tUENj\nvemcHoUVDAAAIOm7oSJz54cKD41SvLO3kuKvUERYlAYnDVdBfrHpPOCi/NPkh3Xs5CF9vf8LORwB\nuv4nt5hO6jEYMAAAgCSpsalBwweO1beFuTpYtEfDBoyVMyJWklQgBgz4nv6JgxUf3VfL1/8fnaoo\nVlhwpK4ZfoMSY/qZTvNrDBgAAKCFO65/xHQC4DEhQaG698Zfy26zqfDkIVXXVjBgdDGOwQAAANp9\naKu++maDbDaeGsC/2Gw2hQaHKTgoVJJUcOKgik4dMVzl31jBAACgh9pz+GsVnDioipozkmXp/hm/\nMZ0EdKnkhEEKDgzR5r0b1ORu1OwJ9yk0KEwOB0+JPYlHEwCAHmjlZ/+piJAoTU37ucJCIkznAN0i\nNDhMKb1TldI7VYWlefrqm09UcCJPg5Ou1g1jbzWd5zcYMAAA6CF2HPxSh4r36nzjOQ3qO0wTRtxo\nOgkwJqX3EKX0HiLpu3cAd7ub1OhulN1mV4Aj0HCdb2PAAACgB8jc+aEOl+zXL2c9YzoF8Dpnz7n0\nzueLZbfbZbM5dMtP71dwYIhsNpvpNJ/EgAEAgJ85V1+nYycPKTwkUo1NDdp3NEdN7kb9YtqvTKcB\nXmnOxH+WLEs2u11f7/9Cb37ymq4bNVORYU5FhcUoKjzGdKJPYcAAAMCPFJ8+oryiPWpsbNC5hnNq\naDyvm665Q+EhkabTAK/lsDuaL48fPlVDk0fqRNkxHSn5VkdKvlVsZILSh01RYkxS8/VY3WgfAwYA\nAD6u6NQRHSjcpeLTRxQYEKRJo29WL2eiQoPDTacBPik2KkGxUQmSpImjfqay6pPatj9TknS09KAm\njfqZ+vZKUUxkvMlMr8WAAQCAj9mZt0nHTuYrwBGkwtI8DegzRBNHzVJYSESLV2IBeEZcVKJmXHun\nJKnCdVonyor01y+XKjGmnyaPma2GxnrZbXZ2pfoeAwYAAD5gy77PdLK8WEGBwQoKCNbsCffpfMN5\nOewOBQUGm84DeoyYyHjFRMZr2ICfaO+R7dr+7UYFBQar6NQRuc5WaNLoWQoMCJZlueV2u9Unrn+P\nOxU0AwYAAIa43U3S9/txF5bmq+jUYVXWlMlut+vEmUJFR/aS2+1WdW25+sSl6KZr7lDI9+9GLH13\nTn8A5lx9xbgWn5+uPKGT5cWqbzgvu92h6tpy7T60RZU1ZQoICFRKYqrOVJUqOiJOQfUxCgoIMVTe\ntRgwAAC4TE1NjWpsalCju1FNTY2qrClTTV21JEuW9f2HLB0+vq/FE4q9h3apV0QfnWk6LFlSavII\nXTv8Bs7BD/io+Og+io/u0+bv1Z0/q7PnXQoLjtCxk4e0KedznW+sU5n7qMqrTyki1KnzDeeUEN1X\nSQlXyGazySbbd7/abJJsCgwIVFL8Fc33Z1luhQSHyW6zd+NX2TGbZVmW6QgAADypqqrKdAIA+CWn\n09nhdbxr3AEAAADg0xgwAAAAAHgMu0gBAAAA8BhWMAAAAAB4DAMGAAAAAI9hwAAAAADgMQwYAAC/\nZ1mWZsyYIbvdrtWrV5vOadPDDz+swYMHKywsTAkJCbr11lv17bffms5qpaKiQk888YSuuuoqhYWF\nqX///nr00UdVXl5uOq1Nr7/+uqZMmaLo6GjZ7XYdO3bMdFKzxYsXa+DAgQoNDVVaWpo2bdpkOqmV\n7OxszZ49W0lJSbLb7Vq2bJnppHa99NJLGjdunJxOpxISEjR79mzt27fPdFYrixYt0qhRo+R0OuV0\nOpWRkaG1a9eazurQSy+9JLvdrieeeKLD6zJgAAD83quvviqHwyFJ379hlfcZN26cli1bpgMHDmj9\n+vWyLEtTp05VY2Oj6bQWSkpKVFJSoldeeUV79+7VihUrlJ2drbvuust0Wpvq6up000036YUXXjCd\n0sK7776rJ598Us8995xyc3OVkZGhGTNmqKioyHRaC7W1tRo5cqQWLlyo0NBQr/33I0lZWVl6/PHH\ntWXLFn3xxRcKCAjQ1KlTVVFRYTqtheTkZL388svatWuXduzYoeuvv1633nqrdu/ebTqtXVu3btWS\nJUs0cuTIzn0PWAAA+LFt27ZZycnJ1qlTpyybzWatXr3adFKn7N6927LZbFZeXp7plA6tXbvWstvt\nlsvlMp3Sru3bt1s2m80qLCw0nWJZlmWlp6db8+bNa7EtNTXVevbZZw0VdSwiIsJatmyZ6YxOq6mp\nsRwOh/XRRx+ZTulQbGys9frrr5vOaFNlZaU1aNAga+PGjdbkyZOtJ554osPbsIIBAPBbLpdLd999\nt5YsWaL4+HjTOZ1WW1urpUuXKjU1VQMHDjSd06GqqioFBwcrLCzMdIpPqK+v186dOzV9+vQW26dP\nn67NmzcbqvI/1dXVcrvdiomJMZ3SrqamJr3zzjs6d+6cJk6caDqnTfPmzdPcuXM1adIkWZ18d4uA\nLm4CAMCYRx55RDNnztSNN95oOqVTFi9erKefflq1tbUaNGiQ1q1bp4AA7/6vurKyUr///e81b948\n2e28btkZZ86cUVNTkxITE1tsT0hIUGlpqaEq/zN//nyNGTNG48ePN53SyjfffKPx48fr/PnzCg0N\n1apVqzR06FDTWa0sWbJER44c0cqVKyV1fhdTfhIAAHzKc889J7vdfsGPrKwsLV++XHv27NHLL78s\nSc2vvHX2Fbjuas3Ozm6+/j333KPc3FxlZWVp2LBhmjFjhlwul1e2SlJNTY1uvvnm5n3Ku8ultKJn\nWbBggTZv3qzVq1d75XEjV155pfbs2aNt27bp8ccf15133qmcnBzTWS0cPHhQv/vd7/TWW281H8Nm\nWVanfobyTt4AAJ9SVlamsrKyC14nOTlZjz76qN58880Wr6o3NTXJbrcrIyOjW56AdrY1NDS01faG\nhgbFxMRo0aJFuv/++7sqsdnFttbU1GjmzJmy2Wxat25dt+4edSmPa05OjtLT01VQUKD+/ft3deIF\n1dfXKzw8XO+8845uu+225u2PPfaY9u/fr8zMTIN17YuMjNSiRYt03333mU65oF//+tdatWqVMjMz\nNWTIENM5nTJt2jQlJSVp6dKlplOavfHGG3rooYeahwvpu5+hNptNDodDtbW1CgwMbPO23r3uCgDA\nP4iLi1NcXFyH13vxxRf11FNPNX9uWZZGjBihV199VbfccktXJjbrbGtb3G63LMuS2+32cFXbLqbV\n5XJpxowZRoYL6fIeV28QFBSksWPHasOGDS0GjE8//VRz5841WOb75s+fr/fee8+nhgvpuyfu3fVv\nvbPmzJmj9PT05s8ty9KDDz6oIUOG6Le//W27w4XEgAEA8FN9+/ZV3759W21PTk7WgAEDuj/oAg4f\nPqz3339f06ZNU69evVRcXKw//OEPCgkJ0axZs0znteByuTR9+nS5XC6tWbNGLpereTeuuLi4Cz7p\nMKG0tFSlpaXKy8uTJO3bt0/l5eVKSUkxevDvggULdO+99yo9PV0ZGRn685//rNLSUj3yyCPGmtpS\nW1ur/Px8Sd8NvYWFhcrNzVVcXJySk5MN17X02GOPacWKFVqzZo2cTmfz8SyRkZEKDw83XPd3zzzz\njGbNmqWkpCS5XC6tXLlSWVlZ+uSTT0yntfDD+3T8WFhYmGJiYjRs2LAL37jLzmkFAICX8dbT1BYV\nFVkzZsywEhISrKCgICs5Odm65557rIMHD5pOayUzM9Oy2WyW3W63bDZb84fdbreysrJM57Xy/PPP\nt2j84VdvON3q4sWLrQEDBljBwcFWWlqa9eWXX5pOauWHv+9//Dt/8MEHTae10tb3pc1ms1544QXT\naS088MADVkpKihUcHGwlJCRY06ZNszZs2GA6q1M6e5pajsEAAAAA4DGcRQoAAACAxzBgAAAAAPAY\nBgwAAAAAHsOAAQAAAMBjGDAAAAAAeAwDBgAAAACPYcAAAAAA4DEMGAAAAAA85v8D44EzgkU82mYA\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a743b38>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "output mean, variance: -0.0003, 2.2534\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Although the shapes of the output are very different, the mean and variance of each are almost the same. This may lead us to reasoning that perhaps we can ignore this problem if the nonlinear equation is 'close to' linear. To test that, we can iterate several times and then compare the results."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "out = h(data)\n",
      "out2 = g(data)\n",
      "\n",
      "for i in range(10):\n",
      "    out = h(out)\n",
      "    out2 = g(out2)\n",
      "print ('linear    output mean, variance: %.4f, %.4f' % \n",
      "       (np.average(out), np.std(out)**2))\n",
      "print ('nonlinear output mean, variance: %.4f, %.4f' % \n",
      "       (np.average(out2), np.std(out2)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "linear    output mean, variance: -0.0171, 7493.0617\n",
        "nonlinear output mean, variance: -1.8226, 26276.2379\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Unfortunately we can see that the nonlinear version is not stable. We have drifted significantly from the mean of 0, and the variance is half an order of magnitude larger. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The Extended Kalman Filter"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The extended Kalman filter (EKF) works by linearizing the system model at each update. For example, consider the problem of tracking a cannonball in flight. Obviously it follows a curved flight path. However, if our update rate is small enough, say 1/10 second, then the trajectory over that time is nearly linear. If we linearize that short segment we will get an answer very close to the actual value, and we can use that value to perform the prediction step of the filter. There are many ways to linearize a set of nonlinear differential equations, and the topic is somewhat beyond the scope of this book. In practice, a Taylor series approximation is frequently used with EKFs, and that is what we will use. \n",
      "\n",
      "\n",
      "Consider the function $f(x)=x^2\u22122x$, which we have plotted below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xs =  np.arange(0,2,0.01)\n",
      "ys = [x**2 - 2*x for x in xs]\n",
      "plt.plot (xs, ys)\n",
      "plt.xlim(1,2)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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YkAxo2igrAAAAt8lqrdbGAyv05Y73VW2tspszmcwaO/BBjR04hYfogdtEWQEA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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a84d198>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We want a linear appoximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation (update) of the Kalman filter we know its current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
      "\n",
      "We can use any linear function that passes through the curve at (1.5,-0.75). For example, consider using f(x)=8x\u221212.75 as the linearization, as in the plot below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def y(x): \n",
      "    return 8*x - 12.75\n",
      "plt.plot (xs, ys)\n",
      "plt.plot ([1.25, 1.75], [y(1.25), y(1.75)])\n",
      "plt.xlim(1,2)\n",
      "plt.ylim([-1.5, 1])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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wZlWAoWQYhipqS1S0/yMVFX80qFO8bLYgTcrOV37e2Zo6do7CQsKHoFOcCsIK\nAAAnaeNuQyv+YK4t/Jp04SyCCjAU+u/i5Q4og7lI3mYL0qSsMzQzb56mjZ2j8NDIIegUp4uwAgDA\nSejqNrRgmeRy9dcykqTHbvVdT8BI4DJcKqveq6LiDdpavGFQtxkmoAQ+wgoAACdh2cvSjlJz7Zm7\npJhIZlUAT3M/SX6XthZv0Nbijwf1oEYCyvBCWAEAYJC2FRta9pK5dt1l0uVzCSqApzidPdpfuUNb\nizdoW8nHauloOuGYIFuwJufka8a4eZo6ZrbCQyOGoFMMBcIKAACD0NPjPv2rx9lfS46XVt7uu56A\n4aK7p0t7DhZpa/EG7SjdqHZH6wnHBAeFaHLOLM0cN09TxhRwkfwwRVgBAGAQVrwibdprrq1dIsXH\nMKsCnApHV4d2lm3S1uIN2lW2SY7uzhOOCQ0O05QxBZoxbp4m5+Rzm+ERgLACAMAJ7Ck39LN15to3\nLpK+fj5BBTgZbR3N2l66UdtKPtaeg0XqcXafcEx4SISmjp2jGePmamL2TB7UOMIQVgAAGIDT6T79\ny9HVX0uIlVYv9l1PQCBpaKnT9tJPtLX4Y5Uc2imX4TrhmMiwaE3rDSjjM2coOCh4CDqFPyKsAAAw\ngCdflzbsMNdWLZKS45lVAY6npuGQthV/rG0lH6u8Zv+gxsRExmlG7lzNGHeWctOnyGa1eblLBALC\nCgAAx1F6yNBPnjHX/uNs6VuX+qYfwF+5H9JYrG0lH2tbySeqaagc1LiEmGTNGHeWZoybq+yU8bJa\nrF7uFIGGsAIAwDEYhqHvPSq1H3XNb0yktPZOyWJhVgVwOntUfGintpV8om2ln6iptX5Q41ITsjQj\nd66mjztT6Ylj+O8JAyKsAABwDM/9VXp7s7m24jYpfTRfrDBydXZ1aHf5Fm0v+UQ7ywrV4Wgb1Ljs\nlPGakXuWpueepaS4NC93ieGEsAIAwOdU1Bi680lz7ZIC6cav+KYfwJea2xq048BGbSv5RHsrtsrp\n7DnhGKvVprz0qZqee6am5Z6pUVEJQ9AphiPCCgAARzEMQ7c8LrW099ciw6Vn7+H0L4wMhmHIfqRS\nO0o/1fbST1Vu3ydDxgnHhQSFalJOvqbnnqkpOQWKCIsagm4x3BFWAAA4ym//Jf19g7n2yM1STipB\nBcOXy3Cp+NBOd0Ap+VSHm6oHNS4yPEZTx8zW9NwzNSFrBs9AgccRVgAA6GWvN7ToCXPtnOnSD670\nTT+AN3UHRt1nAAAgAElEQVR2dWhP+RZ9sO8fOtRQLEdPx6DGJcQma/rYMzU990yNSZ0oK7cYhhcR\nVgAA6HXbSqmhpX89LER6fqlktTKrguGhoeWwdpRu1I4DhdpXuW1Q159IUmZSrvv6k7FnKjUhi1Mi\nMWQIKwAASHrtbUOvv2OuPXCTND6LL2UIXC7DpcraUu04sFE7Sjeq8nDpoMbZrEHKy5ymaWPnaOqY\n2YqLTvRyp8CxEVYAACNefZOhH64w12ZPkhZf7Zt+gNPR1e3Q3oqt2nnAPYPS3NYwqHHhoZGaklOg\nablzNDHrDIWHRni5U+DECCsAgBFv8Sqp9qjvc8FB0rp7paAgZlUQGBpa6rTzQKF2HijUvopt6nZ2\nDWpcYmyKRkdmKzMuT5dfeIVsNr4awr9wRAIARrS/fWjot/8y137yXWnqWIIK/JfL5VR5TbF2lRVq\nx4FCHTp8YFDjLLIoJ3WCpvae3pUSn6FNmzZJEkEFfomjEgAwYjW2GLr5MXNt+jjpnmt90w8wkA5H\nm/YcLNLOA4XaVbZZrR1NgxoXEhymSVkzNWXMbE0ZM0vREaO83CngOYQVAMCIdecaqaquf91mk9Yt\nlUKCmVWB7xmGodqGQ9pZVqidBzappGqXXC7noMbGR4/W1LGzNWXMbI1Ln6rgoGAvdwt4B2EFADAi\nvbXR0Lo3zLU7r5FmTSSowHe6e7q0v3KHdpUVamfZJtU31QxqnEUWZaeM19QxBZo6drZSE7K5vTCG\nBcIKAGDEaW03tHC5uTYxW/rvG3zTD0a2+uYa7SrbrF1lm9wXx/cM7uL4sJAITco+Q1PGFGhSdr6i\nI2K93Ckw9AgrAIAR595npLLq/nWLxf3wx7BQfomG9/U4u1VatVu7yjZpV9lm2Y9UDHps0qg0TRlT\noCljZis3bRIXxWPY4wgHAIwoH2w1tOZ1c+1H35DmTSOowHsaWuq0u3yzdpVt1t6DRXJ0dw5qnM0W\npLz0qZoypkCTc2Zp9KhUL3cK+BfCCgBgxOhwGLrpEckw+mtj06SHFvquJwxP/bMnm7W7fLOq6w8O\nemxsVIKm5MzSlDEFGp8xTaEh4V7sFPBvhBUAwIjxs3XSvs+dcfPsPVJkOLMqOH31zTXaU17Ud+3J\nYGdPrBarxqRO1OScWZqck6+0xBwujgd6EVYAACPCxt2GVvzBXFv4NemiWXwpxKnp6nGouHKndpdv\n1p7yItU0VA56bHTEKE3OztfkMbM0IWuGIkKjvNgpELgIKwCAYa+r29CCZZLL1V/LSJIeu9V3PSHw\nGIahmoZK7S7fot3lW1RSuVPdzsHductisSonZbwm5+RrUna+MpLGymqxerljIPARVgAAw96yl6Ud\npebaM3dJMZHMqmBg7Z2t2luxVbvLt2hveZEaWutOPKhXTEScJmWfoUk5+ZqQNUORYdFe7BQYnjwW\nVhwOh5YsWaJXXnlFHR0duvjii7V27Vqlp6cfd8yvf/1r3XjjjaaaxWJRR0eHQkJCPNUaAGAE21Zs\naNlL5tp1l0mXzyWo4IucLqcO1uzX7vIt2lNepPKa/TIM14kHSrJabRqTOlGTss/ou/aE2RPg9Hgs\nrCxatEh//etf9corryg+Pl533HGHvvKVr2jTpk2yWo//H2pERIQOHDgg46hbsxBUAACe0NPjPv2r\nx9lfS46XVt7uu57gf+qbarTnYJH2lG/Rvopt6uhqH/TYuOjRmpydr0k5ZygvY7rCQyO82Ckw8ngk\nrDQ1NemFF17Qr3/9a1188cWSpN/85jfKzs7WW2+9pfnz5x93rMVi0ejRoz3RBgAAJitekTbtNdfW\n/FiKj2FWZSTrcLRrf+V27TlYpL3lRTrcVH3iQb2CbSHKzZjinj3JzldSXDp37gK8yCNhZdOmTeru\n7jaFkoyMDE2aNEkfffTRgGGlo6NDOTk5cjqdmjlzph588EHNnDnTE20BAEawPeWGfrbOXPvGRdKV\nF/DFcqRxOntUZt+nvQe3ak9FkQ7a98s1yFO7JCk1IUuTss/QxKwzlJs+WcFBnAECDBWPhBW73S6b\nzaaEhARTPTk5WTU1NccdN3HiRL344ouaMWOGmpubtWrVKp199tnaunWrxo0bd9xxhYWFnmgb6MMx\nBW/guPIdp0v6/uoJcnT13w42NrJHN164U4WFPT7szDM4tgZmGIaaOupV3XhA1Y2lqmkuH/RduyQp\nNChCqaNylDYqV2mjxioi1H1hfOvhHm09vM1bbfscxxU8KS8vzyP7GTCs3HfffVq2bNmAO3jnnXdO\n+cPPOussnXXWWX3r8+bN0xlnnKFf/epXWrVq1SnvFwAwsr32/mhtO2B+bsWPrzyohJjADyo4tvau\nFtkby1TddEDVjQfU3tUy6LFWi1VJMZlKHTVWaaPGKj4yhVO7AD8xYFhZvHixrrvuugF3kJmZqZ6e\nHjmdTtXX15tmV+x2u84777xBN2O1WpWfn6/9+/cPuF1BQcGg9wkM5LNfkTim4EkcV75VesjQU383\n175ytvST748N+C+gHFv9OhztKj60Q/sqtmlfxTZV1x88qfHJ8RmamDVTE7NmalzGVIUGh3mpU//H\ncQVvaGpq8sh+BgwrCQkJXzi161hmzZql4OBgvfnmm/rWt74lSaqsrNSePXs0b968QTdjGIa2bt2q\n/Pz8QY8BAOAzhmFo4XKpvbO/FhMpPXWnAj6ojHTdPV06UL23L5wcrDm5606iI0ZpQuYMTchyv0ZF\nnfj7DQDf88g1K7GxsVqwYIHuuusuJSUl9d26eMaMGbrkkkv6trv44ot15pln9p1a9sADD2ju3Lka\nN26cmpubtXr1au3cuVPPPvusJ9oCAIwwz78h/XuTubbiNil9NEEl0LhcTlXUlmhvxTbtr9iu0qrd\nJ3XdSUhQqMamT9bErBmakDlTaYnZBFYgAHnsOStPPPGEgoKCdPXVV6ujo0OXXHKJfvvb35r+h6G0\ntFTZ2dl9601NTVq4cKHsdrtiY2OVn5+v9957j2lIAMBJq6gxtORX5tolBdKNX/FNPzg5LsMle/1B\n7avYrn2V21VSueOknndisViVnZynCVnTNT5zhnJSJig4KNiLHQMYCh4LKyEhIVq9erVWr1593G0O\nHDhgWl+5cqVWrlzpqRYAACOUYRi65XGp5ajvtpHh0jN3c/qXvzIMQ4cbq7SvYrv2V27X/sodau04\nuXPck+My+sJJXsZUhYdGeqlbAL7isbACAICv/O5N6e8bzLVHbpbGpBFU/IVhGKprsmt/5Q7tr9yu\n4sodamo7clL7iItK1PjM6RqfNV3jM6YrNireS90C8BeEFQBAQLPXG7r9l+baOdOlH1zpm37Q70hz\nbd+syf6K7WporTup8ZHhMcpLn+oOKJnTNXpUKjNlwAhDWAEABLTbVkoNRz1SIyxEen6pZLXypXao\nucPJDhVX7tD+Qzt0pLn2pMaHhURoXPoU5WVO0/iM6UpNzJLVYvVStwACAWEFABCwXnvb0OvvmGsP\n3CSNzyKoeJthGDrSXKviQztUXLnzlMJJcFCIxqZOcoeTzOnKTMqVzWrzUscAAhFhBQAQkOqbDP1w\nhbk2e5K0+Grf9DPcuS+Ir1bxoZ0qPrRDJZU7T/q0riBbsMakTlRexlTlZUxTdkqegmzcsQvA8RFW\nAAABafEqqbahfz04SFp3rxQUxKyKJxiGIfuRSpUc2qniQztVcmjnSV8Qb7MFKSdlgvLSpyovc2rv\n7YRDvNQxgOGIsAIACDj/+5Gh3/7LXPvJd6WpYwkqp8rlcupQXVlvMNmlkqpdautoPql9HB1OxmVM\nVU7qeIUEhXqpYwAjAWEFABBQmloN3fyYuTZ9nHTPtb7pJ1B193TpYE2xSqp2qfTQLpVW71HnSTyE\nUXKf1pWTMl7jCCcAvISwAgAIKHeukQ4d7l+32aR1S6WQYGZVBtLhaNeB6j0qObRTJVW7dLCmWD3O\n7pPaR3BQiMakTtS49CkalzFV2cl5nNYFwKsIKwCAgPHWRkPP/9VcW/ItadZEgsrnNbUecc+aVO1W\nSdUuVdWVyzBcJ7WP0JBw5aZOUm76FOWmT1FWci4XxAMYUoQVAEBAaG03tHC5uTYhS7r/Rt/0409c\nhks1Rw6ptDeclFbtVn1zzUnvJzIsujeYTNa49ClKS8zhVsIAfIqwAgAICD95Viqr7l+3WNx3/woL\nHXmzKl09Dh2sKVZp1W4dqNqjA9V71O5oPen9xEUlamz6ZOWmTVZu+mQlx2fwEEYAfoWwAgDwex9s\nNfTka+babf8lzZs2MoJKU9sRlVXvVeGBd1TbXKnfbaiR09Vz0vtJjs/oCya5aZMVH5PkhW4BwHMI\nKwAAv9bhMHTTI5Jh9NfGpEkPf993PXmTy+VUVX1574zJXh2o3nNKp3TZrEHKTM5VbtokjU2brDGp\nExUVHuOFjgHAewgrAAC/9sAL0r4Kc+25e6TI8OExq9Le2aoyuzuUHKjeq3L7Pjm6O096P+EhEcpJ\nnegOJ+mTlZU8jtsIAwh4hBUAgN/auNvQL35vri38mnTRrMAMKi7DJXt9RW842auy6r2qaag8pX0l\nxCRrTNpEjU2dpLFpE5WSkMX1JgCGHcIKAMAvdXUbWrBMch11t92MJOmxW33X08lq62hWmX2fyux7\nVVa9T+U1+0/6wYuS+5SujKSxirDGaXR0ui495yuKjYr3QscA4F8IKwAAv7TsZWlHqbn29J1STKR/\nzqo4nT06VFemcvu+3oCyT4cbq05pX5HhMRqTOlFjUidqbOoEZfae0lVYWChJBBUAIwZhBQDgd7YV\nG1r2krl23WXSl+b5R1AxDEMNLYdVZt+ncvs+ldv3q6K2RN3OrpPel8ViVWpCVm84maCclAkaPSpV\nFot//LMCgC8RVgAAfqWnx336V4+zv5YcL6283Xc9tTtaVVFT4p41qdmvg/b9am5vOKV9RYRFKydl\nfF8wyU4Zr7CQcA93DADDA2EFAOBXVrwibdprrq35sRQfMzQzDd093aqqO6DymmL3rEnNftU2HDql\nfVktVqUl5ignZbxyUicoJ2W8Ro9KY9YEAAaJsAIA8Bt7yw39bJ259o2LpCsv8M6Xe5fLKfuRSh2s\nKdbBmv06WFOsQ3Vlp/TARUmKiYzrmy3JSRmvrORxCg0O83DXADByEFYAAH7B6TS04BHJcdRlH/Ex\n0urFntm/YRiqa7LrYE2xKmqLVV5TrIraEnWdwjNNJCk4KERZSeOUnTK+N5zkaVRUIrMmAOBBhBUA\ngF9Y8yfpo+3m2qpFUnL8yX/5/+wC+IM1xTpYW6KKmmIdrC1Wh6PtlHqzWKxKjc9UVkqespPzlJ2S\np9SEbNmstlPaHwBgcAgrAACfKz1k6N6nzbWvnC1dM//EYz8LJhW1JaqoLemdOSlRW2fLKfcTFz1a\nWcnjek/lylNWUq5CuQgeAIYcYQUA4FOGYWjhcqn9qLOxYiKlp+7UF06pMgxDR5prVVFbosrDpe5Z\nk9oStXU0n/LnR4bHKDs5T1nJ4/qW0RGjTnl/AADPIawAAHzq+Tekf28y137xQyk10VBtQ5UqDx9Q\nRW2xKmpLVVlbqnZH6yl/VlhIhLKScpWZPM49Y5Kcq/joJK4zAQA/RVgBAPhMZa2hJb8y16blHlJL\n5xrd/dQBOU7x4ndJCgkOU8boMcpKGqesZPcrcVSqrBbraXYNABgqhBUAwJBydHXoUF2ZKmoPaNET\nk9TSntP3t+CgDp0x4ec6UF17UvsMCQpVRtJYZSbl9r7GKTkuTVYugAeAgEZYAQB4hWEYam5r0KG6\nA6o8fECHDruXdY3VMmRob/n52rr/ctOYs6b9VjFRAweV0JBwZYweq8zRY3sDCsEEAIYrwgoA4LQ5\nnT2qaahU5eEDqqor06HDZTpUV6bWjqZjbt/eGav3tiww1VITd2v6uH+YahGhUcoYPUaZybnKGJ2r\nzKSxnMoFACMIYQUAcFJa2pvcgaSurG9pP1Ihp3PwT31/d/NCObqi+9ZtNoe+dv5vNXXsLPesSdJY\nZYweq7jo0Vz8DgAjGGEFAHBM3T3dqm2oVFV9eW8oKVfV4TI1tzec1n5LKuappHKeqXb/jS7dd/2y\n09ovAGD4IawAwAj32UMVq+rKe4NJuarry1XTcEgul/O09h0cFKK0hGylj85R+uixigwbpy/dkWva\npmCidM93eOAiAOCLCCsAMIK0djT3hZHq+nJV1R9Udf1BObo6TnvfsZHxSk/MUdroMcoYPUbpiTka\nPSrVdOH7dT83VHvUxExwkLTuXikoiFO9AABfRFgBgGGow9Gm6voKVdeXy36kQtV15ao+UqGW9sbT\n3rfNFqTU+CylJWYrPXGM0hKzlZaYo+iI2AHH/e9Hhn77L3PtJ9+VpuUSVAAAx0ZYAYAA1uFok/1I\npez1B92h5EiFqusPqqm13iP7j4tKVFpijlITs5WWkKX00WOUNCpNNtvJ/d9HU6uhmx8z16blSvdc\n65E2AQDDFGEFAAJAe2erO5QcqXC/esNJo4dCSVhIhFITspSWkN07U5Kt1MRsRYRGeWT/d66RDh3u\nX7fZpBfulUKCmVUBABwfYQUA/IRhGGppb+wLJTVHLU/3DlyfCbIFKzk+Q2kJ2UpNyHIHlMRsjYpK\n9Notgv+v0NDzfzXXlnxLmjWRoAIAGBhhBQCGmNPlVH1TjWoaKlVzpFI1DYd6l5XqcLR55DOsVpuS\nRqUpNSFLKQlZSutdJsamyDaET3pvbTe0cLm5NiFLuv/GIWsBABDACCsA4CUdjjbVNhxSTcMh9/JI\npWobq1TbWHVSD1AciNVi1ei4NKXEZyo1PkspCZlKic9UUlyagmzBHvmM0/GTZ6UDVf3rFov77l9h\nocyqAABOjLACAKfB6exRfXONahuqVNvoDiUlB/equeOIXv6w1WOfY7MGKSkuTcnxGUqJy/S7UHIs\nH24z9ORr5tpt/yXNm0ZQAQAMDmEFAE7AMAw1ttbrcGO1DjdW6XBjVW84qVJdk/20H5x4tJDgMCX1\nzpSkxGUoJSFTyfGZQ3761unqcBhasEwyjP7amDTp4e/7ricAQOAhrACA+i9ud4cRe++y2h1IGqvV\n1ePw6OdFh8cqOT5DyfGZSo5Ld8+YxGcoNipBVovVo5/lCw+8IO2rMNeeu0eKDGdWBQAweIQVACOG\ny3Cpua1BdU12HW6sVl1jtQ43Vfe9d3R3evTzbNYgJY5KUXJchpLi0pUcl963jAjzzC2B/VHhbkO/\n+L259r2vSRfNIqgAAE4OYQXAsOJ09uhIy2HVNdlV12RX/WfBpHe9u6fL458ZExmnpLh0JY1KU1Jc\nuprr2hUTnqDzz744oE7d8oSubkMLHpFcrv5aRpL02A981xMAIHARVgAEnPbOVncQaa5RfVPNUcGk\nRg0th+UyXCfeyUkKD4lQUly6Ro9K0+i4NCWNStXo3nASFhJu2rawsFCSRlxQkaRHfiNtLzHXnr5T\nio1iVgUAcPIIKwD8TndPl44017rDSHOtjjTXqK7JHUzqm+zq6Gr3yueGBoe5w8io1KNe6Ro9KlVR\n4TFee2jicLGt2NDDvzbXrr1M+tI8/r0BAE4NYQXAkOvu6VZDy2E1tBzuDSOfBZMaHWmq9djT2o8l\nPCRCib1BJDE2RYmxn4WSNEVHxBJITlFPj/vuXz1H3RgtOV765e2+6wkAEPgIKwA8ztHdqSPNh9XQ\nUquGljodaa7VkZbD7mVzrZrbGmTIOPGOTlFMZFxvEElRQmyKRsemuANKbIoiwqIJJF6w8lVp015z\nbc2PpfgY/l0DAE4dYQXASXEZLrW0NepI78xIY2udGlrq1NByuDegHFZbZ4tXewi2hSg+NkkJMclK\njE1WQkyKEkel9K6nKCQ41KufD7O95Ybuf95c+68LpSsvIKgAAE4PYQVAH8Mw1NbZooaWur4Q0thS\np4bW/mVT6xE5XT1e7cNisWpUVILiY5KUEOMOJQmxyX1hJDpy1LB4Fslw4HIZuulRyXHUTdbiY6Rf\n3eG7ngAAwwdhBRghXC6nWjqa1NR6RI2tdWpsrVdjS7172Vavpt733U7P39r38yyyKDYqXvHRSYqP\n6X8l9C7johMVZAv2eh84fWtelz7cZq6tWiQlxzOrAgA4fYQVIMAZhqHOrnY1tR1RU+uRo5b1amw9\noqbWejW2HVFLW4NXbul7LFarTXFRiYqLGa346NGKi3Yv4wkjw8qBKkNLnzbXvnK2dM183/QDABh+\nCCuAnzIMQ+2OVjW3Nai5rUFNbUf63je3N6ip9UhfvavHMaS9RYRFKy4qQXG9QSQuOrH35V6PjYyT\ndQQ+Y2QkMQxD33tUau/sr8VESk/dKW5gAADwGMIKMMS6uh1qaW9Uc3ujWtob1NzWqOb2BrX0Lpvb\nG9XS5l72OLuHvL/Q4DCNik7UqKgExUUlalR0Yv+y933o5x6CiJHn+Tekf28y137xQyl9NEEFAOA5\nhBXgNLkMlzocbWptb1JLR5Na2pvU2t6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pU6eqpKTE7pEAwK+e2y7991Fr9uIjUp8IdlUA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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f9149830b00>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is not a good linearization for $f(x)$. It is exact for $x=1.5$, but quickly diverges when $x$ varies by a small amount.\n",
      "\n",
      "A much better approach is to use the slope of the function at the evaluation point as the linearization. We find the slope by taking the first derivative of the function:\n",
      "\n",
      " $$f(x) = x^2 -2x \\\\\n",
      " \\frac{df}{dx} = 2x - 2$$, \n",
      " \n",
      " so the slope at 1.5 is $2*1.5-2=1$. Let's plot that."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def y(x): \n",
      "    return x - 2.25\n",
      "\n",
      "plt.plot(xs, ys)\n",
      "plt.plot([1,2], [y(1),y(2)])\n",
      "plt.xlim(1,2)\n",
      "plt.ylim([-1.5, 1])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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IPdsdOnRIL730kmpqauTxePT888/r4YcfVktLiyorK+/ZrrGxMRbDBpbwTGEr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\nIdW+/H5mWgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f9149fb0668>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we can see that this linearization is much better. It is still exactly correct at $x=1.5$, but the errors are very small as x varies. Compare the tiny error at $x=1.4$ vs the very large error at $x=1.4$ in the previous plot. This does not constitute a formal proof of correctness, but this sort of geometric depiction should be fairly convincing. Certainly it is easy to see that in this case if the line had any other slope the errors would accumulate more quickly. \n",
      "\n",
      "To implement the extended Kalman filter we will leave the linear equations as they are, and use partial derivatives to evaluate the system matrix $\\mathbf{F}$ and the measurement matrix $\\mathbf{H}$ at the state at time t ($\\mathbf{x}_t$). Since $\\mathbf{F}$ also depends on the control input vector $\\mathbf{u}$ we will need to include that term:\n",
      "\n",
      "$$\n",
      "\\begin{aligned}\n",
      "F \n",
      "&\\equiv {\\frac{\\partial{f}}{\\partial{x}}}\\biggr|_{{x_t},{u_t}} \\\\\n",
      "H &\\equiv \\frac{\\partial{h}}{\\partial{x}}\\biggr|_{x_t} \n",
      "\\end{aligned}\n",
      "$$\n",
      "\n",
      "All this means is that at each update step we compute $\\mathbf{F}$ as the partial derivative of our function $f()$ evaluated at  x. \n",
      "\n",
      "We approximate the state transition function $\\mathbf{F}$ by using the Taylor-series expansion \n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "** orphan text\n",
      "This approach has many issues. First, of course, is the fact that the linearization does not produce an exact answer. More importantly, we are not linearizing the actual path, but our filter's estimation of the path. We linearize the estimation because it is statistically likely to be correct; but of course it is not required to be. So if the filter's output is bad that will cause us to linearize an incorrect estimate, which will almost certainly lead to an even worse estimate. In these cases the filter will quickly diverge. This is where the 'black art' of Kalman filter comes in. We are trying to linearize an estimate, and there is no guarantee that the filter will be stable. A vast amount of the literature on Kalman filters is devoted to this problem. Another issue is that we need to linearize the system using analytic methods. It may be difficult or impossible to find an analytic solution to some problems. In other cases we may be able to find the linearization, but the computation is very expensive. **\n",
      "\n",
      "In the next chapter we will spend a lot of time on a new development, the unscented Kalman filter(UKF) which avoids many of these problems. I think that as it becomes better known it will supplant the EKF in most applications, though that is still an open question. Certainly research has shown that the UKF performs at least as well as, and often much better than the EKF. \n",
      "\n",
      "I think the easiest way to understand the EKF is to just start off with an example. Perhaps the reason for some of my mathmatical choices will not be clear, but trust that the end result will be an EKF."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example: Tracking a Flying Airplane"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will start by simulating tracking an airplane by using ground based radar. Radars work by emitting a beam of radio waves and scanning for a return bounce. Anything in the beam's path will reflects some of the signal back to the radar. By timing how long it takes for the reflected signal to get back to the radar the system can compute the *slant distance* - the straight line distance from the radar installation to the object.\n",
      "\n",
      "For this example we want to take the slant range measurement from the radar and compute the horizontal position (distance of aircraft from the radar measured over the ground) and altitude of the aircraft, as in the diagram below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import ekf_internal\n",
      "ekf_internal.show_radar_chart()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w+umFX9veZGXB9OlB96GqVYOe+8ceG149kqRIM/hLKh4uuCDoz//227kX8u7t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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a78f278>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As discussed in the introduction, our measurement model is the nonlinear function $x=\\sqrt{slant^2 - altitude^2}$. Therefore we will need a nonlinear \n",
      "\n",
      "Predict step:\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "\\textbf{Linear} & \\textbf{Nonlinear} \\\\\n",
      "x = Fx & x = \\underline{f(x)} \\\\\n",
      "P = FPF^T + Q & P = FPF^T + Q\n",
      "\\end{array}\n",
      "$$\n",
      "\n",
      "Update step:\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "\\textbf{Linear} & \\textbf{Nonlinear} \\\\\n",
      "K = PH^T(HPH^T + R)^{-1}& K = PH^T(HPH^T + R)^{-1}\\\\\n",
      "x = x + K(z-Hx) & x = x + K(z-\\underline{h(x)}) \\\\\n",
      "P = P(I - KH) & P = P(I - KH)\\\\\n",
      "\\end{array}\n",
      "$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As we can see there are two minor changes to the Kalman filter equations, which I have underlined. The first change replaces the equation $\\mathbf{x} = \\mathbf{Fx}$ with $\\mathbf{x} = f(\\mathbf{x})$. In the Kalman filter, $\\mathbf{Fx}$ is how we compute the new state based on the old state. However, in a nonlinear system we cannot use linear algebra to compute this transition. So instead we hypothesize a nonlinear function $f()$ which performs this function. Likewise, in the Kalman filter we convert the state to a measurement with the linear function $\\mathbf{Hx}$. For the extended Kalman filter we replace this with a nonlinear function $h()$, giving $\\mathbf{z}_x = h(\\mathbf{x})$.\n",
      "\n",
      "The only question left is how do we implement use $f()$ and $h()$ in the Kalman filter if they are nonlinear? We reach for the single tool that we have available for solving nonlinear equations - we linearize them at the point we want to evaluate the system.  For example, consider the function $f(x) = x^2 -2x$\n",
      "\n",
      "\n",
      "The rest of the equations are unchanged, so $f()$ and $h()$ must produce a matrix that approximates the values of the matrices $\\mathbf{F}$ and $\\mathbf{H}$ at the current value for $\\mathbf{x}$. We do this by computing the partial derivatives of the state and measurements functions:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the State Variables\n",
      "\n",
      "So we want to track the position of an aircraft assuming a constant velocity and altitude, and measurements of the slant distance to the aircraft. That means we need 3 state variables - horizontal distance, velocity, and altitude.\n",
      "\n",
      "$$\\mathbf{x} = \\begin{bmatrix}distance \\\\velocity\\\\ altitude\\end{bmatrix}=  \\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the System Model\n",
      "We will model this as a set of differential equations. So we need an equation in the form \n",
      "$$\\dot{\\mathbf{x}} = \\mathbf{Ax} + \\mathbf{w}$$\n",
      "\n",
      "where $\\mathbf{w}$ is the system noise. \n",
      "\n",
      "Let's work out the equation for each of the rows in $\\mathbf{x}.$\n",
      "\n",
      "The first row is $\\dot{x}_{pos}$, which is the velocity of the airplane. So we can say \n",
      "\n",
      "$$\\dot{x}_{pos} = x_{vel}$$\n",
      "\n",
      "The second row is $\\dot{x}_{vel}$, which is the acceleration of the airplane. We assume constant velocity, so the acceleration equals zero. However, we also assume system noise due to things like buffeting winds, errors in control inputs, and so on, so we need to add an error $w_{acc}$ to the term, like so\n",
      "\n",
      "$$\\dot{x}_{vel} = 0 + w_{acc}$$\n",
      "\n",
      "\n",
      "The final row contains $\\dot{x}_{alt}$, which is the rate of change in the altitude. We assume a constant altitude, so this term is 0, but as with acceleration we need to add in a noise term to account for things like wind, air density, and so on. This gives us\n",
      "\n",
      "$$\\dot{x}_{alt} = 0 + w_{alt}$$\n",
      "\n",
      "We turn this into matrix form with the following:\n",
      "\n",
      "$$\\dot{\\mathbf{x}} = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0& 0& 0 \\\\ 0&0&0\\end{bmatrix}\n",
      "\\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix} + \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "Now we have our differential equations for the system we can somehow solve for them to get our familiar Kalman filter state equation\n",
      "\n",
      "$$ \\mathbf{x}=\\mathbf{Fx}$$\n",
      "\n",
      "Solving an arbitrary set of differential equations is beyond the scope of this book, however most Kalman filters are amenable to Taylor-series expansion which I will briefly explain here without proof. \n",
      "\n",
      "Given the partial differential equation \n",
      "\n",
      "$$\\mathbf{F} = \\frac{\\partial f(\\mathbf{x})}{\\partial x}$$\n",
      "\n",
      "the solution is $e^{\\mathbf{F}t}$. This is a standard answer learned in a first year partial differential equations course, and is not intuitively obvious from the material presented so far. However, we can compute the exponential matrix $e^{\\mathbf{F}t}$ using a Taylor-series expansion in the form:\n",
      "\n",
      "$$\\Phi = \\mathbf{I} + \\mathbf{F}\\Delta t + \\frac{(\\mathbf{F}\\Delta t)^2}{2!} +  \\frac{(\\mathbf{F}\\Delta t)^3}{3!} + \\ldots$$\n",
      "\n",
      "You may expand that equation to as many terms as required for accuracy, however many problems only use the first term\n",
      "\n",
      "$$\\Phi \\approx \\mathbf{I} + \\mathbf{F}\\Delta t$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\Phi$ is our system matrix. We cannot use greek symbols in Python, so the code uses the symbol `F` for $\\Phi$. This is admittedly confusing. In the math above $\\mathbf{F}$ represents the system of partial differential equations, and $\\Phi$ is the system matrix. In the Python the partial differential equations are not represented in the code, and the system matrix is `F`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the Measurement Model\n",
      "\n",
      "The measurement function for our filter needs to take the filter state $\\mathbf{x}$ and turn it into a slant range distance. This is nothing more than the pythagorean theorem.\n",
      "\n",
      "$$h(\\mathbf{x}) = \\sqrt{x_{pos}^2 + x_{alt}^2}$$\n",
      "\n",
      "\n",
      "The relationship between the slant distance and the position on the ground is nonlinear due to the square root term.\n",
      "So what we need to do is linearize the measurement function at some point. As we discussed above, the best way to linearize an equation at a point is to find its slope, which we do by taking its derivative.\n",
      "\n",
      "$$\n",
      "\\mathbf{H} \\equiv \\frac{\\partial{h}}{\\partial{x}}\\biggr|_x \n",
      "$$\n",
      "\n",
      "The derivative of a matrix is called a Jacobian, which in general takes the form \n",
      "\n",
      "$$\\frac{\\partial \\mathbf{h}}{\\partial \\mathbf{x}} = \n",
      "\\begin{bmatrix}\n",
      "\\frac{\\partial h_1}{\\partial x_1} & \\frac{\\partial h_1}{\\partial x_2} &\\dots \\\\\n",
      "\\frac{\\partial h_2}{\\partial x_1} & \\frac{\\partial h_2}{\\partial x_2} &\\dots \\\\\n",
      "\\vdots & \\vdots\n",
      "\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "In other words, each element in the matrix is the partial derivative of the function $h$ with respect to the variables $x$. For our problem we have\n",
      "\n",
      "$$\\mathbf{H} = \\begin{bmatrix}\\frac{\\partial h}{\\partial x_{pos}} & \\frac{\\partial h}{\\partial x_{vel}} & \\frac{\\partial h}{\\partial x_{alt}}\\end{bmatrix}$$\n",
      "\n",
      "where $h(x) = \\sqrt{x_{pos}^2 + x_{alt}^2}$ as given above.\n",
      "\n",
      "Solving each in turn:\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{pos}} &= \\\\ &=\\frac{\\partial}{\\partial x_{pos}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ &= \\frac{x_{pos}}{\\sqrt{x^2 + x_{alt}^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "and\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{vel}} &=\\\\\n",
      "&= \\frac{\\partial}{\\partial x_{vel}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ \n",
      "&= 0\n",
      "\\end{aligned}$$\n",
      "\n",
      "and\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{alt}} &=\\\\ &= \\frac{\\partial}{\\partial x_{alt}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ &= \\frac{x_{alt}}{\\sqrt{x_{pos}^2 + x_{alt}^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "giving us \n",
      "\n",
      "$$\\mathbf{H} = \n",
      "\\begin{bmatrix}\n",
      "\\frac{x_{pos}}{\\sqrt{x_{pos}^2 + x_{alt}^2}} & \n",
      "0 &\n",
      "&\n",
      "\\frac{x_{alt}}{\\sqrt{x_{pos}^2 + x_{alt}^2}}\n",
      "\\end{bmatrix}$$\n",
      "\n",
      "This may seem daunting, so step back and recognize that all of this math is just doing something very simple. We have an equation for the slant range to the airplane which is nonlinear. The Kalman filter only works with linear equations, so we need to find a linear equation that approximates $\\mathbf{H}$ As we discussed above, finding the slope of a nonlinear equation at a given point is a good approximation. For the Kalman filter, the 'given point' is the state variable $\\mathbf{x}$ so we need to take the derivative of the slant range with respect to $\\mathbf{x}$. \n",
      "\n",
      "To make this more concrete, let's now write a Python function that computes the Jacobian of $\\mathbf{H}$. The `ExtendedKalmanFilter` class will be using this to generate `ExtendedKalmanFilter.H` at each step of the process."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import sqrt\n",
      "def HJacobian_at(x):\n",
      "    \"\"\" compute Jacobian of H matrix for state x \"\"\"\n",
      "\n",
      "    horiz_dist = x[0]\n",
      "    altitude   = x[2]\n",
      "    denom = sqrt(horiz_dist**2 + altitude**2)\n",
      "    return array ([[horiz_dist/denom, 0., altitude/denom]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, let's provide the code for $h(\\mathbf{x})$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def hx(x):\n",
      "    \"\"\" compute measurement for slant range that would correspond \n",
      "    to state x.\n",
      "    \"\"\"\n",
      "    \n",
      "    return (x[0]**2 + x[2]**2) ** 0.5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now lets write a simulation for our radar."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy.random import randn\n",
      "import math\n",
      "\n",
      "class RadarSim(object):\n",
      "    \"\"\" Simulates the radar signal returns from an object flying \n",
      "    at a constant altityude and velocity in 1D. \n",
      "    \"\"\"\n",
      "    \n",
      "    def __init__(self, dt, pos, vel, alt):\n",
      "        self.pos = pos\n",
      "        self.vel = vel\n",
      "        self.alt = alt\n",
      "        self.dt = dt\n",
      "        \n",
      "    def get_range(self):\n",
      "        \"\"\" Returns slant range to the object. Call once for each\n",
      "        new measurement at dt time from last call.\n",
      "        \"\"\"\n",
      "        \n",
      "        # add some process noise to the system\n",
      "        self.vel = self.vel  + .1*randn()\n",
      "        self.alt = self.alt + .1*randn()\n",
      "        self.pos = self.pos + self.vel*self.dt\n",
      "    \n",
      "        # add measurment noise\n",
      "        err = self.pos * 0.05*randn()\n",
      "        slant_dist = math.sqrt(self.pos**2 + self.alt**2)\n",
      "        \n",
      "        return slant_dist + err"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can implement our filter. I have not yet designed $\\mathbf{R}$ and $\\mathbf{Q}$ which is required to get optimal performance. However, we have already covered a lot of confusing material and I want you to see concrete examples as soon as possible. Therefore I will use 'reasonable' values for $\\mathbf{R}$ and $\\mathbf{Q}$.\n",
      "\n",
      "The `FilterPy` library provides the class `ExtendedKalmanFilter`. It works very similar to the `KalmanFilter` class we have been using, except that it allows you to provide functions that compute the Jacobian of $\\mathbf{H}$ and the function $h(\\mathbf{x})$. We have already written the code for these two functions, so let's just get going.\n",
      "\n",
      "We start by importing the filter and creating it. There are 3 variables in `x` and only 1 measurement. At the same time we will create our radar simulator.\n",
      "\n",
      "    from filterpy.kalman import ExtendedKalmanFilter\n",
      "\n",
      "    rk = ExtendedKalmanFilter(dim_x=3, dim_z=1)\n",
      "    radar = RadarSim(dt, pos=0., vel=100., alt=1000.)\n",
      "    \n",
      "We will initialize the filter near the airplane's actual position\n",
      "\n",
      "    rk.x = array([radar.pos, radar.vel-10, radar.alt+100])\n",
      "    \n",
      "We assign the system matrix using the first term of the Taylor series expansion we computed above.\n",
      "\n",
      "    dt = 0.05\n",
      "    rk.F = eye(3) + array ([[0, 1, 0],\n",
      "                            [0, 0, 0],\n",
      "                            [0, 0, 0]])*dt\n",
      "                            \n",
      "After assigning reasonble values to $\\mathbf{R}$, $\\mathbf{Q}$, and $\\mathbf{P}$ we can run the filter with a simple loop\n",
      "\n",
      "    for i in range(int(20/dt)):\n",
      "        z = radar.get_range()\n",
      "        rk.update(array([z]), HJacobian_at, hx)\n",
      "        rk.predict()\n",
      "                       \n",
      "Putting that all together along with some boilerplate code to save the results and plot them, we get"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from filterpy.kalman import ExtendedKalmanFilter\n",
      "from numpy import eye, array, asarray\n",
      "import book_plots as bp\n",
      "\n",
      "dt = 0.05\n",
      "rk = ExtendedKalmanFilter(dim_x=3, dim_z=1)\n",
      "radar = RadarSim(dt, pos=0., vel=100., alt=1000.)\n",
      "\n",
      "# make an imperfect starting guess\n",
      "rk.x = array([radar.pos-100, radar.vel+100, radar.alt+1000])\n",
      "\n",
      "\n",
      "rk.F = eye(3) + array ([[0, 1, 0],\n",
      "                        [0, 0, 0],\n",
      "                        [0, 0, 0]])*dt\n",
      "\n",
      "rk.R = radar.alt * 0.05 # 5% of distance\n",
      "rk.Q = array([[0, 0, 0],\n",
      "              [0, 1, 0],\n",
      "              [0, 0, 1]]) * 0.001\n",
      "'''\n",
      "wv = .1**2\n",
      "wa = .1**2\n",
      "rk.Q = array([[dt**3 * wv/3, dt**2*wv/2, 0],\n",
      "              [dt**2*wv/2, dt*wv, 0],\n",
      "              [0, 0, wa*dt]])'''\n",
      "rk.P *= 50\n",
      "\n",
      "\n",
      "xs = []\n",
      "track = []\n",
      "for i in range(int(20/dt)):\n",
      "    z = radar.get_range()\n",
      "    track.append((radar.pos, radar.vel, radar.alt))\n",
      "    \n",
      "    rk.update(array([z]), HJacobian_at, hx)\n",
      "    xs.append(rk.x)\n",
      "    rk.predict()\n",
      "\n",
      "\n",
      "xs = asarray(xs)\n",
      "track = asarray(track)\n",
      "time = np.arange(0,len(xs)*dt, dt)\n",
      "\n",
      "plt.figure()\n",
      "bp.plot_track(time, track[:,0])\n",
      "bp.plot_filter(time, xs[:,0])\n",
      "plt.legend(loc=4)\n",
      "plt.xlabel('time (sec)')\n",
      "plt.ylabel('position (m)')\n",
      "\n",
      "\n",
      "plt.figure()\n",
      "bp.plot_track(time, track[:,1])\n",
      "bp.plot_filter(time, xs[:,1])\n",
      "plt.legend(loc=4)\n",
      "plt.xlabel('time (sec)')\n",
      "plt.ylabel('velocity (m/s)')\n",
      "\n",
      "plt.figure()\n",
      "bp.plot_track(time, track[:,2])\n",
      "bp.plot_filter(time, xs[:,2])\n",
      "plt.ylabel('altitude (m)')\n",
      "plt.legend(loc=4)\n",
      "plt.xlabel('time (sec)')\n",
      "plt.ylim((900,1600))\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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VInIr7A5EFy5coKioiP3793PhwgUMBgMhISGkpqaSk5NDSEiII+sUERER6RQ1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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a8899e8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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0Z84cvfTSSzp8+LBiY2P15JNPavz48bb3KC8v12OPPaZPP/1UpaWlSk5O1jvv\nvKOIiAi7z/r16a7aXjShrr366SM6kn/ANh6cMFo3D7zXwEQwGpczoDHheEdjwvGOxuRqfn53uOQc\nPHjQoTf89XNs6hNXLjk/bf9eny3/h23s7emj5yZ/qCbefpd4FVwZ/xFEY8LxjsaE4x2NydX8/O7w\n5Wr1tbxASuo0WN/+9KmKzxRKksoqzurHbam6Pulmg5MBAAAAdc/hhQdQf3l6eGpgjxvt5lZvXiyL\n1fFLBwEAAABXUW3JGThwoJYuXVrjN1yyZIkGDRp0+R3hVP27DZeXh7dtXFRyQsdP5hqYCAAAADBG\ntSUnPj5eY8aMUbt27fT4449r2bJlOnnyZJX9CgsL9f3332vatGlq27atxo4dq/j4+FoNjaqa+Pip\nTVis3VxOwSGD0gAAAADGqbbk/O1vf1NGRobGjBmjWbNmadiwYQoMDFRgYKDat2+vdu3aqXnz5goK\nCtINN9ygOXPm6JZbblFmZqZmzpxZl98B/9UqqLXdOKcgy6AkAAAAgHEuufBAdHS03nrrLb322mta\nu3at0tLSlJGRoYKCAklSy5Yt1blzZ/Xv3199+vSRp6dnnYTGxVUtOZzJAQAAQOPj0Opqnp6eGjJk\niIYMGVLbeXAVWgW1sRtTcgAAANAYsbqaC2kVFGU3PnbyqCoqKwxKAwAAABiDkuNCfL2bqoVfS9vY\nYjEr/+QRAxMBAAAAdY+S42K4LwcAAACNHSXHxbRqSckBAABA40bJcTEXLj5wlJIDAACARoaS42Kq\nXK52nGflAAAAoHFxuOT06dNH7777rk6cOFGbeXCVQgMjZTL98tdaUJyns+WlBiYCAAAA6pbDJae8\nvFxTpkxReHi4xo0bpwULFqiiguWJ6xsvD2+FNA+3mzt6/KAxYQAAAAADOFxyNm7cqB07diglJUWb\nNm3SrbfeqrCwMP3hD39QWlpabWZEDUUEt7UbZ+cfMCgJAAAAUPdqdE9O586d9dJLL+nAgQNauXKl\nbrnlFn3++efq37+/OnTooGeffVZ79+6traxw0IUl5wglBwAAAI3IFS08YDKZNHDgQL3//vs6cOCA\nfvOb32j//v16/vnnFRsbq/79++urr75ydlY4KJKSAwAAgEbsikqO1WrVihUrdO+996p169b64osv\nFB8frzfffFN/+9vfVFJSoltuuUVPPvmks/PCAREt7UvO0YIsmS1mg9IAAAAAdcujJjtv27ZNH3/8\nsT799FMKgbSkAAAgAElEQVQdOXJEoaGh+t3vfqd77rlH3bp1s+03ZcoU/eEPf9D777+vl19+2emh\ncWn+TZurWZPmOnXmpCSp0lyhY4VH1SooyuBkAAAAQO1zuOTEx8dr27Zt8vHx0ejRo3XPPffohhtu\nkJvbxU8GDRo0SO+//77TgqJmIoLbKiNrk218JH8/JQcAAACNgsMlx8/PT++9955uu+02BQQEXHb/\nMWPGaP/+/VcVDlcusuUFJef4ASVpkIGJAAAAgLrhcMn59NNPFRwcrCZNmlx0+5kzZ3T8+HG1bt1a\nktSkSRNFR0c7JSRq7sIV1g7n7TMoCQAAAFC3HF54oG3btvr666+r3b5w4UK1bdu22u2oW1Eh7ezG\ne4/uVHHJSYPSAAAAAHXnilZXu5jKykpnvRWcILh5uMICf7kHx2Ixa33GDwYmAgAAAOqGU0rOyZMn\ntWTJEoWEhDjj7eAEJpNJfeKS7eb+s2O5rFarQYkAAACAunHJkvPcc8/Jzc1N7u7ukqQ777xTbm5u\nVf4JDAzUp59+qokTJ9ZJaDimV6dBcnNzt43zCrN1IGe3gYkAAACA2nfJhQd69eqlBx54QJL0zjvv\n6Prrr1dMTIzdPiaTSU2bNlWvXr108803115S1FizJs3VrW0vbdn3H9vcul0r1C68k4GpAAAAgNp1\nyZIzcuRIjRw5UpJ0+vRp/eEPf1CfPn3qJBic45ouQ+1Kzr4jOw1MAwAAANQ+h5eQnj17di3GQG3p\nENlVJplk1bl7cfIKs1VaViJf76YGJwMAAABqR7UlZ/Xq1ZKkAQMGyGQy2caXM3DgQOckg1P4ePkq\nLChKOQWHbHOH8vaqY+t4A1MBAAAAtafakjN48GCZTCaVlpbKy8tLgwcPvuybmUwmmc1mZ+aDE7QJ\ni7UrOQdzMyk5AAAAcFnVlpwVK1ZIkjw9Pe3GaHjahMboPzuW2cZZuZkGpgEAAABq1yXP5FxqjIYj\nOizWbpyVmymr1SqTyWRQIgAAAKD2OPww0NOnT+vQoUPVbj906JBKSkqcEgrOFRbUWl4e3rbxqdIi\nFZ7KNzARAAAAUHscLjkpKSkaM2ZMtdvHjh2rxx57zCmh4Fzubu6KCu1gN3eQS9YAAADgohwuOd9/\n/73Gjh1b7fZx48YpNTXVKaHgfG1C7R/ieihvj0FJAAAAgNrlcMnJyclRREREtdtDQ0N15MgRp4SC\n80WFtLcb557INigJAAAAULscLjktW7bUjh07qt2+a9cuNW/e3Cmh4HxhgZF247xCSg4AAABck8Ml\n58Ybb9T777+v9evXV9m2bt06vffeexo5cqRTw8F5gpuHy6RfVlM7UXRM5ZVlBiYCAAAAaofDJefZ\nZ59VYGCg+vXrp9GjR+upp57SU089pZtuukn9+vVTYGCg/vKXv9Tow1evXq3Ro0crMjJSbm5umjNn\nTrX73nfffXJzc9Mbb7xhN19WVqaHHnpIwcHB8vPz05gxY7hs7iK8PL3Vwj/YNrbKqvzCHAMTAQAA\nALXD4ZLTqlUrrV+/XnfccYdWrVqlV155Ra+88orWrFmju+66S+np6Ze8Z+diSkpK1L17d82YMUO+\nvr7VPrflyy+/1Pr16xUeHl5ln6lTp2rBggWaN2+e1qxZo+LiYo0aNUoWi6VGWRqD0BZcsgYAAADX\nV+3DQC8mLCxMs2fP1qxZs5Sff+45K8HBwXJzc7gr2RkxYoRGjBghSZo0adJF98nKytLUqVO1fPly\nDR8+3G5bUVGRZs2apdmzZ2vo0KGSpLlz56pNmzZatmyZhg0bdkW5XFVoiwjtytpoG+cVcsYLAAAA\nrueK2onJZJKbm5vc3NyqPfviDJWVlZo4caL+53/+Rx07dqyyfcOGDaqoqLArM5GRkercubPS0tJq\nLVdDFXrB4gPHWGENAAAALqhGZ3L27Nmjp556SkuWLFFJSYkkyc/PTyNGjNCLL76oDh06XOYdamb6\n9OkKCQnRfffdd9Htubm5cnd3V1BQkN18aGio8vLyqn3f9PR0p+ZsKAqLTtuNDxzZ02j/LBoL/n7R\nmHC8ozHheEdjEBMTc/mdquFwydmxY4euvfZalZaWavTo0erUqZMkKSMjQ19//bVSU1O1du1axcXF\nXXGYX1u5cqXmzJmjzZs3281brVanvH9jFOBrXwaLSwtktVpr9WwcAAAAUNccLjlPPPGEfH19lZ6e\nXuWMzb59+9S/f3898cQTWrRokVOCrVq1Sjk5OWrVqpVtzmw26/HHH9eMGTN06NAhhYWFyWw2q6Cg\nwO5sTm5urgYOHFjteyclJTklY0NjtVq1eOsHKi07dxau0lKh9h2jFfirVdfgGs7/hq+xHutoXDje\n0ZhwvKMxKSoquuLXOnxPzpo1azRlypSLXpLWvn17TZkyRatXr77iIBd64IEHtG3bNm3ZskVbtmzR\n5s2bFR4erpSUFC1fvlyS1LNnT3l6eio1NdX2uuzsbGVkZKhfv35Oy+IqTCYTK6wBAADA5Tl8Jqey\nslK+vr7Vbvf19VVlZWWNPrykpER79uyRJFksFmVlZWnz5s0KCgpSVFSUgoPtzzB4enoqLCzMdn1e\nQECAJk+erGnTpikkJESBgYFKSUlRfHy8kpOTa5SlsQhtEaGDubtt44M5u9W5TYKBiQAAAADncvhM\nTs+ePfXBBx+osLCwyrbCwkJ98MEHNT51un79eiUmJioxMVFnz57V9OnTlZiYqOnTpzv8Hm+//bbG\njRun8ePHq3///vL399eiRYu4z6QaUaH2Z+J+3LZUFZUVBqUBAAAAnM/hMznPP/+8kpOT1bFjR91z\nzz22JZ0zMjL00Ucf6eTJk3rvvfdq9OGDBw+u0UM7Dxw4UGXOy8tLM2fO1MyZM2v02Y1VUseBWvjj\nRyqvOCtJKj5TqPTdq9Q3jjNfAAAAcA0Ol5xBgwYpNTVVjz76qN544w27bYmJifr88881aNAgpweE\nczXx8VO/uOu1cvMvC0Ss2Pi1rulyndxMV/ZQVwAAAKA+qdFzcoYMGaKNGzcqJydHWVlZkqQ2bdrY\nrYCG+m9wwk1aveVbWaznzqLlncjWnsPb1LF1vMHJAAAAgKtXo5JzXqtWrSg2DVigf4h6xFyrjZlr\nbHMbM9dScgAAAOASqi05V7oc9KWeT4P645ou19mVnC37/qPbhtwnd/cr6r0AAABAvVHtT7SDBw+u\n8ZuZTCaZzearyYM6EhvZTU18munM2VOSpDNnTykzexvLSQMAAKDBq7bkrFixoi5zoI65u3sovn0f\n/bTje9vcpsy1lBwAAAA0eE49k4OGJSHmWruSs3Xfz7rtuj/Iw93TwFQAAADA1bmiNYP37NmjH3/8\nUSdPnnR2HtShmKhuaurTzDY+U3ZaWbmZBiYCAAAArl6NSs4nn3yiqKgodezYUQMHDtTGjRslSfn5\n+YqJidH8+fNrJSRqh7ubuzpdcHnaoWP7DEoDAAAAOIfDJedf//qX7rrrLnXp0kWvv/66rFarbVtw\ncLA6d+6suXPn1kpI1J6okPZ248OUHAAAADRwDpecF198UUOHDtXSpUt19913V9l+zTXXaMuWLU4N\nh9pHyQEAAICrcbjk7Nq1SzfffHO120NCQnTs2DGnhELdiQxuZzc+duKIyspLDUoDAAAAXD2HS07T\npk11+vTparfv379fLVu2dEoo1B1f7yYKaR5uG1tl1ZHjB40LBAAAAFwlh0vOddddp9mzZ6usrKzK\ntqNHj+qDDz7QDTfc4NRwqBtcsgYAAABX4nDJeeGFF3T06FElJSXpnXfekSR99913evzxx9W1a1eZ\nTCZNnz691oKi9kSFUnIAAADgOhwuObGxsUpLS1OrVq303HPPSZLefPNN/fWvf1VCQoJ+/PFHtWnT\nptaCovZwJgcAAACuxKMmO3fu3Fmpqak6ceKE9u7dK4vFonbt2ikkJKS28qEOXLj4QO6JbJWVl8rb\ny9egRAAAAMCVc/hMzurVq23/HhgYqN69e6tPnz4UHBfg693UfvEBq0V7j+wwMBEAAABw5RwuOYMH\nD1ZUVJRSUlK0bt262swEA8S2jrcb78raZFASAAAA4Oo4XHI++ugj9ejRQ//4xz/Up08ftW/fXk89\n9ZS2bt1am/lQRzq3SbAbU3IAAADQUDlccu68804tWrRIeXl5+uc//6kOHTro9ddfV48ePRQXF6fn\nn39emZmZtZkVtSg2spvc3X65RSv/5FHln8wxMBEAAABwZRwuOec1b95cv/3tb7V06VIdPXpU7777\nrkJDQ/X888+rc+fOtZERdcDby1ftwu3//jI4mwMAAIAGqMYl59eaN2+u8PBwhYeHy9vbW1ar1Vm5\nYIAql6wd2mxQEgAAAODK1bjkmM1mLV26VL/97W8VHBysMWPGaPny5br33nu1Zs2a2siIOnJhydl9\naLNKSosNSgMAAABcGYefk7NixQrNnz9fCxYsUEFBgVq0aKFbb71VEyZM0ODBg+Xu7l6bOVEHwltG\nq0WzYBWeypckVVSWa+22pbqh928MTgYAAAA4zuGSk5ycrGbNmmn06NGaMGGCbrjhBnl41OhZoqjn\nTCaTBsaP1Ddr59jm1mz5TtcljpWnh6eByQAAAADHOXy52ueff668vDzNnTtXN954IwXHRfXter28\nPX1s4+IzhUrfvcrARAAAAEDNOFxybr31Vvn4+Fx+RzRoTbz91Ccu2W5u4do5LCcNAACABuOqVleD\naxqccJPcTL8cGiVnT+m9b/6ikrOnDEwFAAAAOIaSgyqC/EN1wzXj7eaOnTyqL39436BEAAAAgOMo\nObio4b1vU1KnQXZzGzLXaP/RDIMSAQAAAI6h5OCiTCaTJg59UBEto+3mF6z6UBarxZhQAAAAgAMo\nOaiWp4enbh70/+zmDh3bq427eegrAAAA6i9KDi4pJrKr4jv0tZv7z45lBqUBAAAALo+Sg8sa1fcO\nu/HeIzt06kyRQWkAAACAS6Pk4LJCAyMV/qt7cyxWi7btX2dcIAAAAOASKDlwyIWXrG3em2ZQEgAA\nAODSKDlwSI8O/ezGGVmbtDjtY+WdyDYoEQAAAHBxHkYHQMPQKihKoYGRdqUmdf2XSl3/pWIju2nM\ngEmKCmlvYEIAAADgHM7kwGEJHa696Hxm9jbN/PJpnTl7uo4TAQAAAFUZWnJWr16t0aNHKzIyUm5u\nbpozZ45tW2VlpR5//HHFx8fLz89P4eHhuuOOO3T48GG79ygrK9NDDz2k4OBg+fn5acyYMTpy5Ehd\nf5VGYXDCTQppHn7RbWUVZ5V5eGsdJwIAAACqMrTklJSUqHv37poxY4Z8fX1lMpnstm3atEl//vOf\ntWnTJn3zzTc6fPiwhg8fLrPZbNtv6tSpWrBggebNm6c1a9aouLhYo0aNksViMeIrubQmPn568q6/\n6cGb/6JxA++tsj2n4JABqeBqrFarVm/5Tm9//qQWrv1IFiv/WwYAADVj6D05I0aM0IgRIyRJkyZN\nstsWEBCg1NRUu7n33ntPcXFxysjIUFxcnIqKijRr1izNnj1bQ4cOlSTNnTtXbdq00bJlyzRs2LA6\n+R6Nibubu2Kjuik2qpu8PX01b/k/bNsoObhaFotZ81a8a3vg7P6cXWrq20xDe44zOBkAAGhIGtQ9\nOUVF5x5A2aJFC0nShg0bVFFRYVdmIiMj1blzZ6WlscRxbWsV1NpufLQgy6AkaIhKSot18nSBbWyx\nmDU3dYat4Jy3Zst3nM1xwLb96zT736/r6zX/p31HdvJnBgBo1BrM6mrl5eV69NFHNXr0aIWHn7sv\nJDc3V+7u7goKCrLbNzQ0VHl5edW+V3p6eq1mbSwqKsvsxvmFR/Xzuv/I3a3BHFYurz4e67lFWdp5\n9Gdln8iUJIUFRCsuoq8OHt+hfceq3td14lS+Fi//UuHN29V11HqrvPKsjhTu07HiQzKZ3JRz8oCK\nSo/btq/Y+I0Cm4ZpcKdb5efT3MCkdas+Hu9AbeF4R2MQExNzxa9tED+NVlZW6s4771RxcbEWL15s\ndBz8l6eHt/y8A3S67NwZNqusKjpzXIF+YQYnQ310+uxJrT+QqsP/LTfn5RYdVG7RwUu+dm/e5kZf\ncsorz2p7dpoOn8i0KzTVOVGSqyXbPtKwrnfK3zewDhICAFB/1PuSU1lZqYkTJ2rHjh1auXKl7VI1\nSQoLC5PZbFZBQYHd2Zzc3FwNHDiw2vdMSkqq1cyNyYajMdpx4JffJrUI81NSJ/58jXb+N3x1faxb\nrVYtXfe51m5dIrOlUn6+AerW/hq1Ce2gxUs/UPkFZ/8cdbgwU53jYtXU19/Jic85kJOhlZsWKdA/\nWDf0Hi8fL98av0dWbqbStn+v5s1a6prO1ynQP9jh1+afzFH67tUKad5KCbH95WZyU1l5qfIKj6is\n4qyC/EP14eKXlZ2/v0aZzpQXa8Xuz/SniW8ooKnrFh2jjnfACBzvaEzO36pyJep1yamoqNCECRO0\nc+dOrVy5UiEhIXbbe/bsKU9PT6WmpmrixImSpOzsbGVkZKhfv35GRG50WgW1sSs5R49zX05jtj5j\npb77z2e2ccnZU8pLz77EK6ry8Wqih299Sf/89hUdL8qVJJnNlUpd/+VFV/WrCbPFrJ+2f69TpUXq\n3WmwggJCtStrk95f+KLMlkpJ0oGjuzXl5ufk6eHl8Ptu3fezZn37qu0+mCU/z1fP2AEaN/BeNWsS\ncMnXbtrzoz5JnWkrgD9sXChfn6bKPLRVVllr/B3dTG529+MUlxTq8xX/q+HXTNDp0iLFRHaVh7tn\njd8XAICGxNCSU1JSoj179kiSLBaLsrKytHnzZgUFBSk8PFy/+c1vlJ6erkWLFslqtSo399wPPM2b\nN5ePj48CAgI0efJkTZs2TSEhIQoMDFRKSori4+OVnJxs5FdrNMIvWHyAFdYaJ7PFrJyCLH2cOsOh\n/aNbddToa++Wm8lNP25bqsPH9in/ZI4CmwXrjmEPKyI4Wn3jrteitLm216zctEjxHfqpXXinK8po\nsZj1f9+9pq37fpYkLf15vgL9Q2xF6rz9Obv06D9uU6c2CYqL7qkB3Udo+cZvlFOQpa5te6lHTD+V\nlZfqYG6msnIztT8nQxlZm+zew2q1KH33Ku0+vEV3DntYrUPa66s1/6cTp/KV3PNmdYlOlMVq0b//\n85mWrvvC7rWHju116PuEBUapXXgnWaxW5RzPUlBAqEZfe4+a+wXqs+Xv6Oedy237btu/Ttv2r5Mk\nxUZ20/1jp8vdvV7/jgsAgKtislqtNf9VoZOsXLlS11133bkgJpPOR5k0aZKmT5+utm3b2s2fN3v2\nbN19992Szi1I8Nhjj+nTTz9VaWmpkpOT9c477ygiIsLuNb8+3RUQcOnfrMJxR/IP6tVPp9rGLZoF\n67l7PzAwEaS6u5zBYrXoh43faPmGr3W61LFTymMH/FaDE26Sm8l+cUer1Wr3rKyz5aV6+eM/qvBU\nvm0uuHm4nrjj7RqdZTnvX6s+1KrNNb+nLyggVAVF1S9kUlMmk5umjHtWKzcv1vb/Fo+aaBXUWg+M\nfVYBftVffmaxmPXWF08qKzfzottHXDNBI/pMqPFn11dcvoPGhOMdjcnV/Pxu6K/yBg8efMmHdjry\nQE8vLy/NnDlTM2fOdGY0OCikRYTd5TGFp/JVWlYiX++mBidDbbBaraqoLJeXp7dKy0r00dK37C5X\n/LWB8SNVeOq47QyCJN06+HcaGH/jRff/dcGRJB8vX92e/KD+8dV021z+yaNKz1ilvl2vrzaj2WLW\nviM7deT4ARWXFKpT6x7KP5lzRQVHklMLjnTuLM/fFzzj0L7NfANkkVUlpcWSpPCW0Xrw5ufld5l7\nk9zc3HV78kN67bNHZDZXVtm+dN3n6hLdU23CrnzVGgAA6jOuV8BV8fTwVGhgpN1lagdzM9W5TYKB\nqVAb8k/m6N2vn1NB8TH16TJURacLtDNr40X3jW7VUTcPnCyTyU0bM9do75Gd6taul7pE96zRZ3Zs\nHa9+Xa9X2vbvbXNpO76vtuScLS/VzC+ftrtBf/mGry77Oe5uHurXdZi2H1hvd+aoJkwyadzAezWg\n+wgtXfeFlq7/QlYHn1XTzDdAvxlyn1Zv/U57s7fLv0kLjRkwSUkdzy2gkp1/QKdLixQb2c3hy8xa\nBUXppn536us1s6tss1gt+vDbVzRl3LMKC4xy+DsCANBQUHJw1dq26mhXcvYf3UnJcUGfLfu77f6V\nn3Z8f9F9Av1D1CY0RrcO/p3c3NwlST07DlTPjtWvdng5yUm32JWcrNxMHck/oIjgtlX2/e4/n112\nBTIvTx/9btSTOpibqbLyUoUFRalDRFcF+gfrxr63a/fhLfpoyVu2hQguxiSTwoKi1LZVR4W3bCs/\nX39FBrdVSItzl8mO7DtR/k1b6PMf/vey3y8ypJ1+N+pJtWgWrB4x/VR0+oSaNW1udzlfVMiVLZ99\nXeJYBTcPV+6JbFmtFi1O+9i2reh0gV6a+5C8PX0UGhilXp0GKTF2wGUXSgAAoCGg5OCqtY+Is/sh\ndO+RnQamgTOVVZzVoby9Ol1apL1HdlS7X3jLaP3+pqdrtGyyo1oGhKljVLx2H95im0vb/r1+M+T3\ndvsdyT+o1Q5cknZ78oPq2DpeHVvHV9nWxMdPCTHXqkWzYL05f1qV7X+89UVJUnjLNmri7XfJz+nf\nfbgKinO1fMPX1e7Ts+NATRw6RV6e3ra5S91rcyW6teutbu16S5KKTp/Qmq3f2W0/93e8R4fy9uhf\nqz5UoH+IkjoO0sg+E2xFFQCAhoaSg6vWPryL3TgrN1MVleVyc3OXOz8kNVg5BYf0v18/r8LTl3/w\n5MShU2ql4JzXr9swu5KTnrFSyUk3q0WzlrJardp+YL0Wrv3IbunkixkYf6MSY/tf9vOiw2LVJizW\n7sb9pI6D1CEirka5R197j6JCOig7/4CSOg5QeWW55i55S2fLzyi51y0a3OOmKvci1aZbBk2WxWrR\nj9uWVLvPieJjSl3/hbw8vTWs1611lu1CeSeyNTd1hvJPHlV0WEeFt2yt0rISeXn46Ppet6hZk+aG\nZQMA1H+UHFy1QP8QtWgWbLuXodJcoUf/cZtMJjc19wuSr1cTubt7qEt0oob3Hs/StQ3AqTNFem/h\nCw4VnPgOfWv9BvZu7XrLzzfAtoJbafkZPTf7Pg3uMUoZh7bo6PGDVV7zu5ueUlhglD5d9ncdzN2t\nnrEDNHbAJIc/c2Sfifrfr5+XVVZ5eXjr+iv4gd9kMikxtr9dsfqfSe9WWUmurri5ueu2IffJv0lz\nLV33+SVLYeq6L9Sr0yC1aFZ75bU6p84U6d2vn9OJ//5/yq6sjdr1q/u/dh7coD/d/qa8PX3qPBsA\noGHgp004RfvwLkrfvcpuzmq1qPBUvgr/Oz58bJ+aeDfTkMTRdR8QDjNbzPrn4ld0ovjYZfd1M7np\nxr6313omD3dPDeoxSt/+9IltzmIxa8XGby66f9e2vWyXaD1864tXVCo6t0nQg7f8RfuP7lRc2yS1\nCnLeDfpGFJxff/aIPhN0XeIYnS0vVWl5iTbvSdPmPWk6WvDLw3zLK8s048unddewqSo5e0pWq0Vd\nonte0fLdNXGiOF8fLXnTVnAu5tjJo1q49qMqlyzWVEVlubLz9yvIP0z+TTkzBACuhJIDp2gfUbXk\nXMzmvWkNtuRUmisa3JPirVarDuTsVu6JQ4qJ7Kbg5q0uul/+yRztO7JTTX2b6dSZk9qfs+ui+7m7\neeju4Y9o9eZvVVxSqOF9JtTZ6lzJSTfr8LF92rrvP5fcL75DX91x/R/t5q60VMREdlVMZNcrem19\n5+3lK28vXwUoUMOvGa/h14zXyk2LtGD1P237nCg+phlfPmUbh7aI1P1jn1Ggf8hVfbbVatXB3Ex5\nuHsoIrit3ExuOnWmSN/+9In+s3O5LBbzZd9jzdbvtGbrd2oTFqsOgT0V0aK9w59vtpi1YfdqLfzx\nIxWXFMrLw1sThj6gpE6DruZrAQDqEUoOnKJ9RJfL7yQpO3+/zBZzg7pXx2yu1D+/e007D6QrNqq7\n/t9NT8rLw/vyLzTYoYIMpX48R7knDkuS3N099P9ufEJxbX95gFxG1mZ9vXb2RS/3Oq9deGf17jxE\nOQWHFN+hrzpExCkh5trajl+Fu5u77hmeove++Ysys7fZbTPJpB4x/TSs160XXXUNjhkYP1Lrdv1Q\n7Qp1eYXZeuvzJ3T/2GcU3jL6ovuYzZXan5Ohg7mZKq8oVULMtXb7VlSW65/fvqqdBzdIOrd8dlBA\nmA4d23vRcuPl4a1enYeo6HSBth9YX2V7Vm6msnIzFRkYK0vTU4qL7qmm/32OkNli1qbMtSo+U6he\nnYbodGmxFq6do8zsraqoLLe9R3llmT5OnaHS8jO6tuswuwUXLFaLDuftVeGp42rerKXCAqPk4+V7\n2T/L8w4f26czZ0+rfUQXmc2V2rZ/nby9fNWpdYI8Pex/aXK6tFh5J7LVOrSDKirLtXrLtyo+c1K9\nOw9RdFisw58JAJBMVqvVanSIunA1T0zF5VmtVj33f7+3XWJiMrnpnuEpigxupxc+esBu3wHdR+rM\n2VNKiL1W3dv3MSJujaze8q2+XPmBbTyiz0SNuGa8gYku77PFH+infd9Wmffy8NZDt7ygiOBoLU77\nRCs2Vr/yl3TuzM0zk9415L6M6lRUlmvt1iXKPXFYleYKNfcLUu8u1yn0v8s34+pk5e7RjC+fUqW5\notp9PD28NKD7SJ0tP6OmPs2UGDtAEcHROnm6QDO/fNq21Lh07ph75LZXFBHc9tylkN++qu2/ekDs\npUSGtNOUcc+pqU8zSdL+o7s044unZFX1/9ny9PBS37hkhQZGae3Wf9stb+8IL08feXp4yd3krubN\nWqqw+JhOlf7y3w93dw/dNvi+Sz6Q9rwfNi3UV6tnXXSbf9MWui5xjAbFj5K7u4f2ZG/Th4tfUWlZ\nSbaWiZAAACAASURBVJV9TTJpaNLNGnHN+Fq/XBD1X3r6uQcwJyUlXWZPoOG7mp/fKTlwmoyszfps\n2d9ltpp188DJtput31v4gnYcSK+yv8nkppTbXq2XT10/XpSrn3cuV8uAMC35+XMVFNs/9X7GH78y\n9L6KS9mYuVaz//26U96rf7fhuu26PzjlvdBw/H/27js8qjJ94P53SqYlk55J75CQhF6kSZNmLytr\nQVdxVd61oqw/X3UtWNfedVdd10V9UVTUXTsoICAoPZRAIKT33iaZyWRm3j+yOcuQEAIJJIT7c11z\nkTn1mXBy5txPuZ/KulK2Za5nf94OahoruzU+Ky48mYamWqrqyjqsiwiOY/EVz/DRj6+z7cD6Yx4r\nxC+cWeMuZ9yQaR26iP6y+we+WP8eLQ5b9z9QL1Or1Nxx+eMkdpFtr6GpjiXv3ezRYtSZ1LgxjBsy\njY9+evOYn8nPJ4hZYy5j0tA5EuycwSTIEWcSCXK6QYKcvvPdb8v57tePOl03PnUm18y+4xSXqGt2\nh40n37+N2saqo25z1++fJiFiyCksVefa/3zbA66M3O2889VTXU5k2V39sRVH9A2X28XnP/+Ddenf\nHnvjHtB7GTh3/JVMG3lhl+PfXC4njtYWiqvyWf7Tmx4JE7pLo9YycvAkAswh/Lh1xQmXOSEihWvn\nLCLYL8xj+Ve/fMCqHhy3K77eAcwa8zumjDgfFVBRV4q/T5BkmztDSJAjziQ9eX6XMTnipIsNHXTU\ndb9l/MRV59zSr9JK78/b2WWAA7Bl/9o+D3JySvbz/vcv0Wy3khwzAr2Xgc371nikBVar1PzxgnvJ\nLT141Ae5AJ9grjv3bvQ6I698+gD2/9Ymzxh1sQQ4Ami7ji6fdjO+pgC+PizDXU8E+YZy++8eo76p\nhhaHHa3Gi2hLosfEqEctj1qDXmckPjyZe695ie9Wf0FxbTb5tfupO8bfLsCYpClcd+5ipXIgJXYU\na3f8h0PF+2iyNXS6jwpVp93ksov38bcvH2Pxlc+g1XhRVVdGZV0p63advICw3lrD5+ve5fN176Lz\nMtDisKFRa4kLS2LKiPO7NReUEEIMdP3nyVIMWNGWowc5AGt3fk14UDRJ0cP7RfayQ8UZx9xmx4EN\nXD7txm6Vd0/2FnYc/IWk6GGMT53ZG0XE7XazbNXrSje6HQd/6XS7a+bcyfDECQxLGI+vyZ+vN35I\nS6tdWT9i0ESunnkbJoMPAPdd8wobdn9HoG8ok4fN7ZWyioFBpVIx56zfE+Abwte/fIgbN2FBMTRY\nayg6InGFTqvnznlP8vZXT1JvrelwrEBzCLf/7jGC/EIJ8gvtUbnUKjWhfrGE+sWy4JK72H5gHZkF\nu6htqKTV2cq4IdPIKtqr/I0kx4zgmjl3enQ3bc+i53K7sDbXAyrqrFUcKNhNs9363+QJsUdtnamo\nLebhd2/ssmva8MTxaDU6po28kEDfEP725WOdJvxIjR2Nw+mgrLqQScPmMG3EBfyw5TN+2f19p8dv\n7+LmdLVyqDiDQ8UZaNRaRgyaQLPdyqqtn+NotTMhdaYk5RBCnFGku5o4JR5698Zj1rAmx4zglksf\nQa1Sn6JSde6Fj/+PvLKDx9zunNGXcOmUG7rc5uedX7Pi538o768/98+MSZ7S4zLmlR7gheX3drnN\nWQnncu1FnuNpqurL+HHL51TWlTIuZTrjhkzvt2OLxOnB7XazcsunfLNpmbJs3vSFTB1xPgcL9/D2\nV09ib2lW1gX5hnLH5Y/3OA314Y7VfcftdpOZn47D2UJa3BiP7GnHw+ly8q9vnyP9GGnMj3TR5OuY\nPfZ3Hsua7I28//1LZORuw0urIz58CGcPO5cRgyZ2+jdZb61l9fYv2bDrO4+Kis74GP2475qX+dd3\nz5NVtBdoa4kalzKdCyZeQ4A5+LjKL/oX6a4mziQyJqcbJMjpW88uW3zUtLSHWzTvyS4H8/aUy+1i\nffq35JRkMm7INI90ygAtDjv3/n1+t+bpABiROIEZoy/16LrmcjnJLNjFbxmr2X7EIOuIoFj+32te\n7nFgseLnf/Dzzq87XadWqRkZM4OhURPlS1CcMgcL95CetZH48BSPQN7W0kxe6QGKKnNQqzSclToD\nk96nV899Kh/63G43BeWHsDua+WztO8fM3hbsF8a98186atppq60BvZeh263Y9dYa3v/+xQ5p1LvL\nS6NjxuhLmHvWFR1SWIvTgwQ54kzSk+d3zZIlS5b0cnn6Jbv9fzVfBoMMzjzV7A4b+/N3HnM7H6Mv\nQ2JGnrRybN63huWr/0ZJVT47szYyevBkZU4NgOySffyW8VOn+w5LOAubvUkZswJt84b8mvEjpdUF\nDEs4C1Qq/vbvx/j+t+WdPvw0NNeRFD28R7XYTpeTZT++7pGJSavxYmjCOM5Kncm86TdjcLbN3h4R\nEXHC5xHieAT5WkiNG0NEcKzHcq3Gi2C/MOLDhxAXnnxSsoIVFxcDp+Z6V6lU+PkEEuQbyqjBkzlY\nsJs6azXQVsEQ6Gsh2pJIUvRwxg6Zxu+m3oS30XzU4+m0+uNqWdLrjIwZMg2dVk9DUy2DotJYcN6f\n0XnpySnJPOb+LreTQ8UZ5JcdZOTgSafVnGWizam83oXoaz15fpcxOeKUmJg2m3U7v6a6oQKz0Y+E\niJROu3zsyd7KJWcvOGnl+PawLjVOZytrtv+HK2feoizLLt7XYR+1WkNEcCxXzbyV2sYqXvnsLx1S\nve44+AtqlZpxKTPIzE/vsgzr0r9h0Am2VjXZG/n853dpaKpVlhl0Jp68+V8eD48FlJzQ8YUQ3edj\n9OXuK56mvLYEnVaHvzn4lAQNGrWG2eMuZ/a4y5VlF02+jkNFGd3qaguwP38n//j6aW668D5JRy2E\nGJAkyBGnhFFv4sHr3yS/7BCR/63p9VqjJ6d4v8ccNGU1hezL20FoQBSBvr2f2aumsdLj/b687R7v\nDx0R5Fw18zbGDZmOVqNFpVJhNvlz0wX3sfT7F7AekYVp24H13ZoDZFfWr9Q2VuHvE3RcZd+TvYUP\nVr7cYbLAEYMmykOKEH1Eo9ESHhTd18VAo9aw4Lx7eP2LhzvMVXThpGvxNpj59tePPCpI9uVt5/0f\nXuKG8+454XFKQgjRX/XtCG9xRtFqvEiIGIJeZ0SvM3Ld3Lt55Ia3SIoa5rHd3758lEffW8gvu3/o\n1fMf/uXezmpvpKGplvKaYlwuJ7lHdPdIiBiCl9bLYwzNkNiRPHT937h61u3KTOxHkxwzgnuuep6w\nwP89BLncrqN+tpZWO5v3rWF/XlvXvgMFu/j4pzd4+v+7i7e/erLT2dDHJk/tsgxCiDNDkF8o9179\nEhMOy+I4KGooM0dfyuRhc7nvmlcID4rx2Cc9axOfr3uX03l4bnfHUAohzizSkiP6XFrCuA6DaN24\n+eqXD5iQNqvXun8UVeR2WGZvaeYv7ywA2lK32lqalHUmgxlLQGSnxzIZfJiYNoukqGG89Ol9nabJ\n9fUO4JZLHkat1jBlxPl8uuYtZd3G3T8wZ9zvPQb+VtdX8MbnD1NR19bVbEjsKA4U7OryC3xYwlkk\nRQ/v8nMLIc4cRr2J+bPv4Jwxl1LXWE1CRKoyD5nZ5Mftv3uMV1c8SFl1obLPuvRvsbU0c+U5t3Ko\naC9fbfyA6vpyRgyawJxxvz/mGEJHq4Mft64gr+wgY5OnMnbItJP6GdsVVeSyYt0/yC3JJCV2FPOm\nL5TMcUIIhQQ5os8NjR/HF+v+2WF5k72RgvJDxIUl9cp5jpzL40gZR3Rdiw5JOGY66yC/UK6dvYg3\nv1zSYd2YpClKF5Czhkznq18+UIKohuY60rM2Kg8DVfVlvLbiIarry5X99+ft6PScKlRMHDqbMclT\nGRSZJimghRAdhAVGe7QgtzOb/Ln10kd48ZP7PNL6b963hs371nhsu3HPKn7LWMO0kRdy3vgr0XeS\nIc7ldvHhypeVeYgycrfhaG1h4tDZvfyJ/qeitoSNe35g7Y6vcbpaAdidvZlDxfuYMeoiUuPGEBWS\nIPdGIc5wEuSIPhfiH05cWDK5pR0zA2Xm7+y9IKci57i2P7Jbx9EMiR3JiEETSc/a5LH88NpMvc7I\n+NRzPNI+r9v1LWOHTMPldvHu1894BDhHo9Pquf68P7dlchNCiBMQYA7hlkse4pXP/tJpF9jDOV2t\nbfPz7P6e+PBk4sOHMG3khXgbzDTZG/ls7TsdJiP+ZM1bGPXe2B02fk7/GrfLxYzRlxxzXq6Gplrq\nrNWEBUZ3mlK7rrGaFT//g51ZGzvdv8nWwDeblvHNpmUMjhrG/Fm393iyWSHE6UuCHNEvXDvnTv69\nYSm7szd7LM/MT2fuWVf0yjmKKo8zyDkiFW5XLpvyR/blblcm6YsMiScqJMFjm7OHn+cR5OSWZLI7\nezNqlbpbcwhdPet2knuYfloIIQAiguNYfMUzvP3VU1TUFh9z+xaHjcz8dDLz0/n+t+UE+YZS31SD\no7Wlw7ZOVyv//PZZj2UfrnyF9KxNzJ91u0fa/na7Dv3GBytfxt7SzKCoofzpkofQafXK+q37f+bT\nNW/RfFiX4q4cLNzNX/+/RVwz+05GDZ7UrX2EEAOLJB4Q/YIlIJKbL3qAJTe87bE8pyTTY7b0E+Vo\nbaGspui49onoZksOQKBvCDdf9ABRlgQSI1L5w5y7OtRYhgZEdpgDaNmq1/j3hqVdHlvnZeDxG//J\nxLRZEuAIIXpNaGAUf77yWSakzvQIKCwBkUweOpcAn6OPb6mqL+s0wOnK7uzNvPHFEprtnoHKoaIM\nln73gnKvzyrcw9rt/wHausN9vfFD3v/hpU4DHG+jLzGhgzvNDtfisPH+Dy+SX5bVrfK53W7KaopY\nn/4t69K/7ZXvHiFE35GWHNGvBPpaCPELVwbfO12tZBXtJS2+ZzM7l1YXHHcGnrDjCHKgLZPavTEv\ndrnNBRPntyUTcLuAttnOj0xFffNFD/DJmreU/vKXnr0AP5/A4yqLEEJ0h8ngw/zZd3DVrNuobajC\n1mIlLCgGtUqN3WHj+9+Wsz79W6WV+mi8jb7EhSWxN2drl9sVVmTzxPu3cu74K7G3NJNfltVp97NV\nW1dgtTWwM2sTNQ0VHdZHhSQwZcT5jB48Gb3OSFV9GXuyt7Atc71H12ens61V6d6rX8Rk8DlquQrK\ns/nox9c9WtV/2f09i694ptOxSEKI/k+zZMmSJX1diFOhJzOmilOrrLqQ/PL/1byZTf6kxI7q0TH3\n5e7o0BWuK0F+ocwcc1mPztkZf58g1GoNBwp2dbo+ITyFS86+nrHJ0wj0DWH6yIsZnXz2cZ9HZsQW\nZxK53ntOpVJh1HtjNvkrrdBajZYhMSOZPupiRgyahM5LT37ZQdz8L910gDmEc0ZfwjWz72DS0DlE\nhsRhtTVQXV8BuNsyVKpUOA4LklocNjJyt5FZkE5pdUGn5XG6WsktzfTIeNlWJi8umbKA+bNuIyZ0\nkDJ2x6T3IS4siQlps/A2mD3mQGu2W9mXt53EyFTMJj+P47lcTn7d+xP//PZZao+YR62xuY7N+9dS\nVVdGbmkmUSHxaDVelFYXsCd7M7uzN1PbWIWfdyA6Lz2nilzv4kzSk+d3ackR/U5yzAg27P5eeb83\newuXTbmhR5lyCsoPebyfe9YVDIkZSWVdCSs3f6a0HLWLCOr+eJzjNWvs7zhQsKvTQGf6qIsA8PX2\nZ+qIC05aGYQQort0XnqiLQlEWxIYnzKDrZk/Y9KbSY4ZQZTFMwvl8MQJDE+cQLPdiqO1BV/vAOwO\nG6+veIi8soM9KofZ6MdNFz1AfHjyUbdRqVRMG3khFbUlrEv/RlleVJnLcx/9mWvnLGJ00tnUNlax\nLv1bNmespr6p4xQA7eoaq1i/61sAftj8CWaTf4c511SoiA1PYmjcWMalzOiQxrreWsOW/WtpaW1B\n76UnMSKV2F5KqCOEODoJckS/Mzh6GBqNFqezLTVoRV0JeWUHe5Rl7cjMbbGhg0mMTCUxMpW8siwq\ndnkGOeEnMchRq9TcdOH9rFj7Dlv2r1W6rkUExzE8cfxJO68QQvRUZEg8kSHxx9zOqPfGqPcGQO9l\n4P+55CFe/ewvR2258TUFcM6YS9i4eyXlRyRC8NLqGJ44gYsm/YFA35BulfOSsxeQV3aQvNIDyrJW\np4N/ffc8//ruedRqzQlNItrZpNJu3OSWZJJbksl3m5czKCINH6Mv3kYzgb4Wftz6BY3NdR77XDrl\nBs4ZfclRz1NnrWbnwY1oNV6MGzL9lLYUCTFQSJAj+h2T3oe0uLHsOvSrsmzr/rUnHOS0OOwd5sg5\nvBYt2pLYYZ+I48isdiIMOiPXzLmTueOvYMu+tTicDqaOOL/TwbNCCHG68zH6svjKZ9md/RulVQXU\nNFRiMngT6BtKfPgQYsMGo1apSYoezj+/eZY6azXJ0SMYnXQ2wxLOOu5xMV5aL2659GE+Wf0W2w+s\n77C+swBHo9ZyxTl/IiokgReX36vMwXM8nM5WMgvSj7ndl+vfo7q+nNCASLZmrqOytoSRgyczbeSF\n7MzayMotn9HisAGwYdd3/OnSh/Hzbhub2dJqo7Ami9KNmcSHJ5MSO0q+O4TohMrtdruPvdnpr67u\nf7Uofn5+XWwp+oP0rF9595unlffeRl+euPGfyszdx+NQ0V5e+ewvyvtgvzAeXvB35X1hRTbPLlvs\nsc/9177a7Xly+qOtW9sG/44d27OEDUKcDuR6H1jaH0t6azLPX/f+xMc/vaG0mh9JrzMyctAkZoy6\niIjgOADSszax7MfXcbS20Op0dNgnMSKVQF8LxZW5x5xourdo1FpQofRyaBdoDiE5ZiSBviE0260Y\n9T6MGjypbTzUEVwuJ4eKMyiuzCM+fAgxoYNOSdn7K7fbTWNzHQ1NtbQ6W9FqtJhNAXgbzcecDFyc\nGj15fpeWHNEvpcaNwaj3ViaqszbXs2H394xOmkKrswU/78Bu11zlHtZdASDuiP7c4YEx6L0M2P9b\na6bzMmDxlwGdQgjRF3oruGk3IW0mep2Rpd+/4NGCY/GPYPqoizkrZUaH7mAjBk1keOIEVCoVLreL\nVVtWsGHXdwT5hjJ3/BUeyXBqGirZnf0bq7f/u8tJnVUqNe6jBFrdcbSWpeqGCjbtXeWx7NtfP2Jw\n1FAs/hEEmEPwNvpSXJnDzoObPMYgDYoaypTh55MWN+aUdYlrsjVSUpVPtCXxpJ7T7XZTWJHNnpyt\nFFVkE21JZMboS9ift5O9OVspqy6ktKaQpiMynAKo1RrMJn98Tf74eQcyNGEc41POOaGKVtF3pCVH\n9Fsf//QmG/es7HRdjGUQCy9+EF9v/2Me592vnyb9sK5v86YvZOqI8z22Wb39S/69filu3Fxy9vUn\nJbPaqSQ12+JMIte76I680gNs3LMKs8mPYQnjiQkd1KsBldPZSkbediprS3Hj5mDhbjJyt2PUmbjk\n7OuZOHQ2Bwp28fZXTyld0foLL62O6JBEEiNTmTnmsi7TbfdESVU+r614iMbmOrwNZi6cdC3jU89B\nrdZQb61paxmryEWt1hAbNhg/70BaHDbs/321OOw4Xa0YdCYqaovZceAX6qzVjEo6m4lpsyivKaLJ\n3vjfZA8/U/zfVjZ/7yAumHAtRr33CbfQ6HVGgv3CgLZWNa3WS1p7ekin06FWd/077MnzuwQ5ot/K\nKdnPS5/cd9T1kSHx3Hn5E8rg1sM5na1oNFrcbjcPvftH6q3/q7m656rnO22ir6orA9rSR5/u5KFP\nnEnkehf9VavTgdvtxkurU5bVNFSyae8q9uXtoLKulMSIVLQaL/Zkb6al1U5oYBRjk6cxaehsftj8\nqUeWuHY+Bn/M3r5U1pbicB7fpKzH4u8TxJ+vfK7H87M5XU7ySg+g9zISHhyD2+Xi+eX/R1FFTodt\ntRqvTrsF9gZ/7yBuOPf/JS5ycK+3EooT53a7sdls6PV61Oq2ebmq6ysoqy5gf/4OZo75HSH+4RLk\ndIcEOaenb3/9iJWbPz1qX+rEyDT+dPGDHoNSf975NV9v/BCT3ofLp9/EP77+39geL42OZ29ZNuCb\nnOWhT5xJ5HoXA4G9pRlbSzO+3gEeD+PN9iZanS3ovYyoVCq2b9+ORq1l7NixNNkbySs9SGFFDk22\nelwuF/vzd1JSlX/U82g0WlwuV5dd52Isg7hz3pPd7k7W4rCzM2sje3K20OpsxeVspbAiR+ka11nq\n7VPlmpmLOCttugQ4/ZDb7WbPoW18tPq1DhkIfzf1RqaPukjG5IiB6/wJVzNq8Nl8s+lDMgt2YW9p\n9lh/qGgvb3y5hD9d8hAmvQ81DZV8sf49XC4ndofNI8ABiA5NHPABjhBCiNOPXmfsNIucUW8CTMp7\njfp/32EmvQ8psaM8xgi53C7ySg9SWVdKY3MdZdUFVNaW4usTyIjEiaTGjcZLqyO/LIutmetIP7iR\nmiMmQs0vz+Lt/zzBH+beTUurXTlWVEiCR1Ke2sYqft75NRt3/0DzERO3Hq6vAhyVSk2gb6gEOP2U\nSqXC7XZ1CHAAMnK3KXMHnih52hP9XnhQNDddeD9ut5uWVjtvfP6Ix7w3uSWZvPOfp7hj3hPszt7c\n5dwHgyKHnooiCyGEEH1CrVITH57c5aSpADGhg4gJHcRlU26gur6cf3z9V49McQcKd/PQu3/0PLZa\nwx/mLCIiOI6ftn3Btsz1J5RqGyAldjRFlTlKd3Kj3psQv3DCg2JodToorMih1elA72VA52VA56VH\n52VAo1JT31SLy+UkJnQwTpeDnVm/0my3EhkcR+h/s8oF+4czPvUcfPTHHrsr+s7RxjVlFe2lxWHv\n0bElyBGnDZVK9d9J5R7kzS+WUFB+SFl3qDiDzPx09mRvPur+Wo0Xk4fNPRVFFUIIIU4LKpWKIL9Q\n7pz3FC9/el+XXd1cLidLv3+xR+dTq9ScP+Z6poyeg1ar5dffNlFRWc6UydMICeneZK9H+v2MP+F2\nu1ChRqVSeQxmt9n6V5IH0Tm1Sk2AOYQA3xCiguNJiRuNpofzP0mQI0473gYzt//uMf725WMeLTrr\n0r/hYOGeo+539vDzCDAHn4oiCiGEEP2O0+nk4MGDZGRk0NzcTEtLCy0tLTQ3N9PQ0EBRaTMtga34\nBB//42Fzo53iA7U0WW34BOhxtLSyfXUWVSX1hET5ExoVQGCIP/t35vJK7heoVCp0Oh12+/9q62Ni\nYggODsbb2xuNRkNFRQX5+fm0tLSg1WoJCgoiNDQUi8VCUFAQbrcbq9VKUVERhYWFlJaWotfrGTFi\nBBqNhvz8fD744AOmTp3am79G0YvCgmJ45Ia38PcJ7nFQcyQJcsRpyaj35sJJ1/L65w8py/bmbD3q\n9jqtntljf3cqiiaEEEIcN6vVSl1dHS6XC6fTidPp9PjZ7XYTEhJCa2srOTk57Nmzh9zcXKqqqvDx\naUv5nJeXR319PV5eXh4vtVrNgQMH2LlzJ1artctyqNUqJpyfwsjpiWi0ahrrbDTWNBEW13m2NZdd\nzcEtZfz89VbstrZMb5GRkfj6+tJqVWPQG6ktbaIsr61bmkajITo6mpKSEux2O0OGDCEsLIzffvuN\n/Px88vM7b0my2+1Yrdajrm/X1NTEpk2bPN6Lk2ft2rWcc845fPzxx1xxxRXHvX9JYRmffvopQUFB\n2O12SktLycjIYPbs2fz5z3/uUdkkyBGnrUFRaQT4BHcYMNmZ2ePmYTa19ct1uVyUlZVhtVppbm6m\nqamJ5uZmmpub8fb2JikpidDQgT9Q0e1243K50Gh6t+ZECHHqtLS0UFJSQnh4ODqdrstt3W43VVVV\nFBYWUlxcTFNTEw6HA4fDQX19PTt37uTgwYNKDb/dbsdkMpGUlERsbCwBAQHU1NRw6NAhsrOzKSws\nJCEhgWHDhlFeXk5paSk+Pj74+/vj5+en/BsVFcWIESMYNmzYMct4OnC73coDd2Njo/JvTU0NRUVF\nVFZW0tjYiNPpxGAwkJOTw5YtWygpKaGhoYG4uDiGDx+OzWajvLxceXX3YbxtsPaJJ8aNjo5m+PDh\n+Pn5odPp0Ol06PV6/Pz8CA4OVoKOltYWfvv1V1ZuXYXJ5I8pPIAmfYlynABzCJdOuYERieNRqzU4\nnU6Kiorw8vIiPDy8w3kbGhqoqqoiPDwcvV6Pw+HAarXi79/23dzS0kJhYSFVVVU0NTXhdDoJDAwk\nJiYGo9GIw+GgsrKS8vJyysrKqK6uRq1WYzAYiIyMJCoqioiICOVabg+mAgICTvh31V8da26Zdu+9\n9x7XX3/9SS5NzxQVFfH44493WO7l5dXjIEdSSIteU1lZydq1a8nNzcVms9Hc3IzVamX//v1kZGSQ\nlJTE7NmziYiIQK1WU1hYSG5uLnl5eTgcDoYPH87o0aMZNWoUGo2GAwcOeLzy8vIIDg4mOjoatVqN\n0+nEHOfCL7bz8tRlqdF5a2htVtFY7MLhaKWuro4dO3Z4XA+diYqKYu7cuQQFBVFfX09dXR319fUe\nP9vtduXmXFhYiNvtJjY2lqCgIPR6PQEBAYSGhjJs2DDGjRuH3W5Xbs5lZWWUl5eTnZ3Nrl27qK6u\nxtvbW3n5+Pjg4+ODn58fQUFBpKamMnLkSCIiIggODlau4aqqKiorK2lqaqKiooKCggIKCgpIT08n\nKCiIadOmsWvXLjZv3kxdXR0Oh0O54e/du5eGhgbCw8NJTk5m8uTJjBgxgtjYWFJTUzGZTB6/E5fL\nRU1NDRUVFVRUVFBTU0NjYyM6nY74+Hiio6MJDg7u9s23P3I4HOzbt4/4+HjMZnO393O73eTl5bFv\n3z4cDgd6vR6dTodWq8Vms1FTU0NOTg5Wq5WUlBSGDRtGUlLSUR/4XC4Xhw4dYteuXWRkZJCfn09p\naSlNTU1YrVaamprw8vLCYrFgsVgIDQ0lJSWFCRMmYLFY8PLywmaz4XQ6CQ0NPa3/T7qjv6aQEaxU\nawAAIABJREFUbu8aZLVaGTx4MDqdjpKSEnx8fAgODkalUuF0OqmtraWqqorq6mrKy8vJyclRXq2t\nrYwdOxaz2UxGRgZWqxWdTkdpaSlZWVnk5+crlRXx8fGYTCYlmLFarajVatRqNRqNhrq6Oo+uQaea\nr68vF110EREREVitVkJCQoiOjsbhcGCz2fD19cXlcrF7926qq6uJjo4mJiaGmJgYIiMjCQoK4rff\nfuP777+ntLSU+vp6tFotJpOJ8PBwgoKClHtUZWUllZWVVFRUoNfriY2NJTQ0lICAAAICAjCbzVRX\nV1NZWYnRaMTX1xez2YxWq6W6upqqqiqqqqqoq6vzCGSsVitWqxWn8+gJbk6UXq/H398frVar/J9p\nNBrlZ4Dy8nKqq6uJiIhgypQpDBo0iJCQEBobG3G5XMTExBAYGKgErw6Hg5aWFlpbW4mLi2PUqFEn\nPO7F5Xaxdsd/2JO9hcTINGaNuazTbHD9jc1mw2Aw9HUxetWyZcs83r/11lv8+uuvvPfeex7LJ02a\nRFxc3EktS09bcg4dOsQHH3xAVVUVer0ei8XC4MGDGT16NLGxsafvPDnr1q3j+eefZ/v27RQXF3ca\ncS5ZsoR33nmHmpoaxo8fzxtvvEFqaqqy3m63c8899/Dxxx/T3NzMzJkzefPNN4mMjPQ4jgQ5J6ax\nsZH9+/fjdrvx8/PDz88PvV5PVlYW+/btIyMjQ/k3KyurR7VLJyIg1Idr75/ZYbm13sZ7S1bidnVe\nnqCgIPz8/DAajRiNRkwmE0ajkdraWjIzM6mt7Zt0l92l0WjQarUn7YFFp9Mxfvx4pU90RUUFVVVV\nx/xi12g0hIaGEhYWRnh4OCEhITQ0NFBbW4vJZMLf35+oqCiSk5O54IILCA4Oxul0dhgo2pnW1lay\nsrLIy8vDZrNhMpkYOnQoYWFhx2x1c7vdlJeXk5eXpzy8HPkqLy9ny5YtNDY2otfrmTNnDhEREQBU\nVFQowWltbS1Op5PW1lbl39bWVlyuo8850RmtVsuIESOYM2cOoaGhVFZWkpOTw4EDB9i7d2+vdbEw\nGAxKLXz7l4fb7Wbv3r20tLQQGBhIQkICQ4cOZejQoaSkpGA09v8Hl8P9+uuvlJSU4OXlxf79+yks\nLMRsNhMQEEBgYCA1NTWsWbOGqqoqQkNDlUDA29tbqSmOj48HUFp12ytqDq+hz8rKQqvVkpqailqt\nprS0FLVajU6no6ioiIKCAuVacLlc1NXVHXXQc3tlQ21tbY/um2q1GovFQllZWbeO096yEhkZiY+P\nj9KdyWg0MnToUNLS0vDx8VFq9+vq6sjMzKSkpITq6mrMZjOJiYkkJiYSGRnJ/v372bdvH2FhYURE\nRNDU1ERtbS11dXXU1dVRW1tLdnY227Zt48CBAyf8OfsbnU6nVEi1V075+fkRGRlJaGgoZrMZtVpN\nc3MzFouF8ePHk5CQgMlkIjMzk4yMDMxms1JRYbFYMJvN3epBsGnTJry8vPpdUN9fDcQg50gLFixg\n+fLlNDc3d7md1WrF27vjBOo90dMg51j/P6ftPDlWq5Xhw4dz/fXXc91113X4437mmWd48cUXWbp0\nKUlJSTz22GPMnj2bzMxMpf/pXXfdxX/+8x8+/vhjAgMDWbx4MRdeeCHbtm0b8LWXh3O5XBQUFFBY\nWEhdXR1qtZqgoCBSUlJobGzkwIEDuFwu9Ho9ra2t2O127HY7LS0tuFwuZeDevn372Lp1KxUVFahU\nKiorK7v9BazX65XWAJPJhMFgwGAwkJCQQEpKCunp6axfv15pUYiKiiI2Npa4uDhUKhU7d+5k+/bt\n7Ny5E4CkpCSPV2xsLJWVlRQXFwMoNVzbSr6l3lGhlEPlVjMiYgYff/RHbDab8hDS/kU+bNiwDkHw\nkb/L9PR0Vq9ejcPhwNfXFz8/P3x9fT1+1ul0ygNvdHQ00NYfur22tLq6mqKiIjZv3kx6erryhRYa\nGqoMnIyKimL48OGEh4crtfSH1xrW1tZSVlZGeno6e/bsUWooGxoacDqd+Pn5YbFYlIe16OhopWa0\nvLyc+vp6kpOTmTp1qlLDX1NTg8PhIDU1lcDAQIqKiti5cye//PILBw8eJDs7m71797J+/foOvxs/\nPz9CQkIICQkhKCgIs9mM1WolJyeHoqIiqqurKS4uVv6PuqLRaIiIiKC4uBiDwcCIESNITEwkODiY\n+Ph4EhMTaWxsJCsri1WrVrFp06ZOgzqDwaAE4L6+vtTW1lJSUoJer2/rE97a1oLX0NBwzDJBW1eO\nwsJCvvrqq25t385isTBs2DBMJpPyt+VwODAajZjNZuLi4jAYDOzbt489e/Zw6NAhtm3bxrZt2zo9\nXmRkJMOHD2fo0KHEx8cTHh6Oj48PJpMJk8lES0sLFRUVlJeXU1JSwvbt29m6dSv19fU4HA7lS6Oy\nspLMzMxOz9EZtVrNoEGDGDp0KKNGjWLMmDGMGTOGwMBADh06RE5ODsXFxTQ2NuJ2u6msrKSoqAiN\nRoPZbMZsNuPl5UV2djaVlZWkpaURGxtLcXExLS0tREdH4+XlRUVFBTqdjvDwcOUVFhaGwWBg7dq1\nbN68mfr6euVvwcvLi8GDBxMdHY2vry9r165l+fLlVFRUnJRa9a788ssv3d42OjoaPz8/srKycDgc\nhIWFUV9fT01N2/gElUqlBGOBgYEEBQURFxdHfHw88fHxuN1utmzZgs1mIyUlhcDAQOx2OyEhIQwa\nNIi4uDh0Oh3Nzc3k5uYqfyNBQUH4+PjgdruVMR3tLcTHa+LEiV1+vtmzZ3frOAcPHuSbb75RusGV\nlpZSWFiIXq/HYDDQ0NBAa2sraWlpWCwWCgsLyc/PJy8vj5KSEioqKkhISODSSy8lJSUFs9mMy+Wi\nsbFRuf8EBQURHBxMcHCwcp+y2+3k5ORQWVlJTU0NNTU11NfXExgYSEhICDabjfr6ehoaGnA4HMr/\nQ1BQEP7+/h6BTPvPXl5ex/17bNf+N3WienJuceZoD3z279/PHXfcwc8//8zo0aNZs2YNu3bt4qWX\nXmLdunUUFxfj4+PDrFmzePbZZ5XnmXZ1dXU88cQTrFixguLiYoKDg5k2bRrPPfecUhF4JIfDwfz5\n8/nuu+/497//zcyZHSujT4V+013NbDbzxhtvcN111wFtNa8RERHceeed3H///UBbtGexWHj++edZ\nuHAhdXV1WCwW/vWvf3H11VcDbd2GYmNj+e6775gzZ45y/NO9JcdutytN79XV1bS0tCi1yBkZGbzz\nzjvk5OT0+nm9vLxITk5Gp9MpNXPNzc3Ex8eTkpKivFJTU0lKSuqT2pKq+jL+vX4ptpYmhiacxbgh\n0zDqe7emor+x2+04HI6jPrD0tPtOdXU1v/76KzqdTglqgoODj9mfvr1LXklJifJQ4uvri7+/P01N\nTVRXV1NYWMimTZv48ccfaW09vvkVYmNjGTx4MEajkZqaGnbv3n3Mroft/P39iY+PVx58OnsNHTqU\nyMhIiouL+fHHH7FarbhcLoKDg5UWqoCAALRaLVqtVmlR02g0x/3gYbVa2bBhAz/++CM2m42goCBi\nYmIYNGgQaWlpBAUFHdfxjqa+vp6CggJqa2spLi5WatOHDh2Kt7c3lZWVHDhwgD179rBnzx4OHDjQ\nadCg0WhOeTBxPEJDQxk+fDhDhgwhNjYWq9VKTU0N1dXVeHl5MXXqVOLi4igrK8Nms6FSqWhsbKSy\nspLc3Fzy8/PRaDQYjUYMBoNHK29719PBgwfT0tJCRkYGarWasLAwoK31Jzw8nNjYWAwGg9IyaTKZ\nlO+bthnm3Wg0GtxuNxUVFWg0Gvz9/WVcnDgu/bV7Zn91prbkLFiwgI8++oi4uDjGjx/PlClT0Gq1\n3HDDDbz44ousWLGCc889l4iICLKysvj73/9OYGAge/bsUVrzrVYrkydPZu/evdxwww2MHTuWyspK\nvvvuO5566immTJnSoSXHbrczb9481q9fzzfffMPkyZO7LPuAbcnpSk5ODmVlZR6BisFgYOrUqWzc\nuJGFCxeybds2HA6HxzZRUVGkpKSwceNGj+WHe+qpp8jOziY/P5/KykpCQkIIDAxUBgsaDAaCg4NJ\nTU0lLS2NtLQ0EhISlP90u92OWq1Gq9V6tD45HA6la095eTlarZb4+HgCAwPRarV4eXl12MflclFa\nWsqOHTtYtWoVOTk5mEwmHA6H0j2mrKyM+vr6Y/7OgoODSUxMJCAgAJfLpTzQmEwmkpOT0ev12O12\ntFoter1eGTeg0WgwGAxERESQkJDA2LFjiY2Nxe12ExAQ0O8Higb5hvLHC+7t62KcUu3/fydLYGAg\n559//nHvp9frldakY2kfhxAVFUV9fT3p6ekUFhZSXl5OVlYWOTk5+Pr6EhkZyZQpU5g5cyaBgZ7Z\nfdxuN83NzUr3mPr6evz8/IiIiMButytZhsxmM/7+/t1OJhEREaFUuJws3t7ezJ07l7lzT+7cTb6+\nvqSlpXV7e7vdTmZmJunp6Wzfvl15NTY2Ehsby6BBg4iMjFS64wQGBiq1eQ0NDTQ0NGC324mNjSUw\nMJBdu3ZRWlpKREQEOp2OgoICnE4nwcHByqD59ldpaSkNDQ2MGTOGmTNnEhISotSgNzc3c/DgQUpK\nSqitrSUhIYHrrruOkSNHsmPHDlQq1Sl76DuRWsnDexaoVCosFktvFkkI0QtOdsKhU92u4HA4uOii\ni3j++ec9lt9yyy0sXrzYY9nFF1/M5MmT+fzzz7nmmmsAeO6559i1axeffvopl19+ubLtAw880On5\nmpqauOSSS9i+fTurVq1i3LhxvfyJjk+/DXJKS0uBttq5w1ksFqUrTGlpKRqNpkONZ2hoKGVlZUc9\n9l/+8pcTKpO3t7fS1QvavrT8/f0xmUzKoPTuOHwwYXt//u7s4+fnR0BAgJIRpf0YZrOZOXPmMGHC\nhA41gi6XC5VKddx/uO0pGgsKCo5rP9G/tNf49Wft3Xb8/f2VRA7nnHNOh+2ys7PJzs7u8lgajUbp\nnnm48vLyXirtmaO9lfaaa67B5XLR0tJyQrWhCQkJx7W90+k8asvG6NGjOyzbuXOncn87Ha53IXqL\nXO/d097Ceqa69dZbOyw7fNxlY2MjdrudwYMH4+/vz/bt25Ug57PPPmPo0KEeAc7R1NfXc+6555KZ\nmcmaNWsYPnx4t8rX0NDAnj1Hn+Nw8ODB3TpOZ/ptkNOVnkba1157LZGRkYSHh+Pv76/0z/X398fb\n21vp697+UJWdna2kHAaUL2Cn00l1dTXV1dXA/4Ke9uwtra2tFBcXY7ValWCmvW/04d0+AgICiImJ\nYcyYMSQnJ9PS0qKMqWnvp+3r63tCn/tMGpckhDg52tO0ngrSdUsI0Vf6yQiOXqNWqzvNrlZTU8N9\n993HZ599plQ0tju8e9ihQ4e47LLLunWuxYsX09zczPbt2xk2bFiPyt1b+m2Q097XuaysjKioKGV5\nWVmZsi4sLAyn00lVVZVHa05paWmXs9t+8MEHx10et9tNTU0Ner0ek8mESqVScrbX19crAcmxgor2\nQaAOhwOXy4VarT7tMhmJ/k/6bIsziVzv4kwi1/vxOVqGwzOBTqfr9Ln0iiuuYOPGjdxzzz2MGjVK\nmTLhqquu8sgUejyV65deeikff/wxTz75JMuWLet2JbvZbO7yWu7uuNvO9NsgJz4+nrCwMFauXKlk\nIbHZbGzYsEHpWzhmzBi8vLxYuXKlR+KB/fv3M2nSpF4tj0ql6jAeoH3Cq84mverqOO2DloUQQggh\nhDgZOmuZqqmp4aeffuLRRx/loYceUpbbbDalZ1K7xMREdu/e3a1zXXjhhZx//vlce+21eHt78+67\n7/as8L2gz1NIHzx4EGgbO5KXl8fOnTsJCgoiOjqau+66i6eeeoohQ4YwePBgnnjiCcxmM/Pnzwfa\nsizceOON3HvvvVgsFiWF9IgRI5g1a1ZffjQhhBBCCCFOic5aXTpb1t4l+Mi53V566aUOQdG8efN4\n9NFH+eyzz5g3b94xy3DVVVdhtVq5+eab8fHx4ZVXXjmej9Dr+jTI2bJlizLAWKVS8cgjj/DII4+w\nYMEC/vnPf3LvvffS3NzMbbfdRk1NDRMmTGDlypUeExm9/PLLaLVarrzySpqbm5k1axYffvjhSc+Q\nIYQQQgghRH/QWatNZ8t8fX2ZPn06zz77LC0tLcTExLBhwwbWrVtHUFCQxz7/93//x4oVK7j66qtZ\nuXIlo0ePpra2lu+//57HHnus06EhN954I42Njdx99934+Pjw5JNP9u4HPQ59GuRMnz79mLOEtwc+\nR6PT6Xj11Vd59dVXe7t4QgghhBBC9GudZdHtKrPusmXLWLRoEW+99RYOh4Np06axevVqZs2a5bGP\nyWRi3bp1LFmyhM8//5ylS5cSGhrKtGnTSEpK8jjX4RYtWkRDQwMPP/wwZrOZ++67rxc/bff1m8lA\nT7bTfTJQIY6HDEwVZxK53sWZRK7343MmTAZ6OjuZk4FKfmEhhBBCCCHEgCJBjhBCCCGEEGJAkSBH\nCCGEEEIIMaBIkCOEEEIIIYQYUCTIEUIIIYQQQgwoEuQIIYQQQgghBhQJcoQQQgghhBADigQ5Qggh\nhBBCiAFFghwhhBBCCCHEgCJBjhBCCCGEEGJAkSBHCCGEEEIIMaBIkCOEEEIIIYQYUCTIEUIIIYQQ\nYgDIzc1FrVazdOlSZdm//vUv1Go1+fn5fViyU0+CHCGEEEIIIU4T7UFLZ6877rgDlUqFSqXq8hjL\nli3jlVdeOUUl7hvavi6AEEIIIYQQ4vg8+uijJCYmeixLTk5mxYoVaLVdP+IvW7aMvXv3smjRopNZ\nxD4lQY4QQgghhBCnmblz53LWWWed8P7Hau05Ec3NzRiNxl4/7omQ7mpCCCGEEEIMAJ2NyTnS9OnT\n+fbbb5Vt21/t3G43r732GsOGDcNoNBIaGspNN91EVVWVx3Hi4uI477zz+Omnnxg/fjxGo5Fnn332\npH224yUtOUIIIYQQ4ox35yuXntTjv7roy149Xm1tLZWVlZ2u66qV5sEHH+Tee++lsLCQl19+ucP6\nW265hX/+858sWLCAO++8k/z8fF577TU2b97Mli1b0Ov1yjmysrL4/e9/z8KFC7n55puJiYnpnQ/X\nCyTIEUIIIYQQ4jRz7rnnerxXqVTs2rXrmPvNmjWLiIgIamtrmT9/vse6jRs38vbbb/PBBx9wzTXX\neJxrypQpvP/++9x8881AW4vPoUOH+M9//sOFF17YC5+od0mQI4QQQgghxGnmtddeIyUlxWOZwWDo\n0TE/+eQTfHx8mDNnjkcrUXJyMhaLhTVr1ihBDkB0dHS/DHBAghwhhBBCCCFOO+PGjeuQeCA3N7dH\nxzxw4ACNjY2EhoZ2ur6iosLjfUJCQo/OdzJJkCOEEEIIIc54vT1m5nTkcrkICgpi+fLlna4PCAjw\neN9fMql1RoIcIYQQQgghziBHS0yQmJjIjz/+yPjx4/H29j7FpepdkkJaCCGEEEKIM4i3tzc1NTUd\nll911VW4XC4ee+yxDuucTie1tbWnoni9QlpyhBBCCCGEOIOMGzeOTz75hLvuuouzzjoLtVrNVVdd\nxZQpU7jtttt47rnn2LVrF3PmzEGv15OVlcWKFSt4/PHHue666/q6+N0iQY4QQgghhBCnka7mwenO\n9rfeeiu7d+/mww8/5LXXXgPaWnGgLWvb6NGj+fvf/86DDz6IVqslNjaWK6+8knPOOeeEy3Cqqdxu\nt7uvC3Eq1NXVKT/7+fn1YUmEOPm2bt0KwNixY/u4JEKcfHK9izOJXO/Hx2az9Titsjh5jvX/05Pn\ndxmTI4QQQgghhBhQJMgRQgghhBBCDCgS5AghhBBCCCEGFAlyhBBCCCGEEAOKBDlCCCGEEEKIAUWC\nHCGEEEIIIcSAIkGOEEIIIYQYsM6Q2VJOOyf7/0WCHCGEEEIIMSDpdDpsNpsEOv2M2+3GZrOh0+lO\n2jm0J+3IQgghhBBC9CG1Wo1er8dut/d1UcQR9Ho9avXJa2+RIEcIIYQQQgxYarUag8HQ18UQp5h0\nVxNCCCGEEEIMKBLkCCGEEEIIIQaUfh3ktLa28sADD5CQkIDRaCQhIYGHHnoIp9Ppsd2SJUuIjIzE\nZDIxY8YMMjIy+qjEQgghhBBCiL7Wr4Ocp556irfeeovXXnuNzMxMXnnlFd58803++te/Kts888wz\nvPjii7z++uts2bIFi8XC7NmzaWxs7MOSCyGEEEIIIfpKv048sGXLFi6++GIuuOACAGJiYrjwwgv5\n7bffgLb0cy+//DL3338/l112GQBLly7FYrGwbNkyFi5c2GdlF0IIIYQQQvSNft2Sc95557F69Woy\nMzMByMjIYM2aNUrQk5OTQ1lZGXPmzFH2MRgMTJ06lY0bN/ZJmYUQQgghhBB9q1+35Nx6660UFhaS\nkpKCVqultbWVBx98kD/96U8AlJaWAhAaGuqxn8Viobi4+KjHraurO3mFFqIfGDx4MCDXujgzyPUu\nziRyvQvRPf06yHn11Vd57733+Pjjj0lLS2PHjh0sWrSIuLg4/vjHP3a5r0qlOkWlFEIIIYQQQvQn\n/TrIefLJJ3nwwQe54oorAEhLSyMvL4+//vWv/PGPfyQsLAyAsrIyoqKilP3KysqUdUIIIYQQQogz\nS78OctxuN2q157AhtVqN2+0GID4+nrCwMFauXMmYMWMAsNlsbNiwgeeff95jPz8/v1NTaCGEEEII\nIUSf6tdBzqWXXsrTTz9NfHw8qamp7Nixg5deeonrr78eaOuSdtddd/HUU08xZMgQBg8ezBNPPIHZ\nbGb+/Pl9XHohhBBCCCFEX+jXQc5LL72Er68vt912G2VlZYSHh7Nw4UIefvhhZZt7772X5uZmbrvt\nNmpqapgwYQIrV67E29u7D0suhBBCCCGE6Csqd3vfLyGEEEIIIYQYAPr1PDm95c033yQ+Ph6j0cjY\nsWPZsGFDXxdJiF63ZMkS1Gq1xysiIqKviyVEr1i3bh0XX3wxUVFRqNVqli5d2mGbJUuWEBkZiclk\nYsaMGWRkZPRBSYXouWNd7wsWLOhwv580aVIflVaInvnrX//KuHHj8PPzw2KxcPHFF7N3794O2x3v\nPX7ABznLly/nrrvu4sEHH2Tnzp1MmjSJ8847j4KCgr4umhC9bsiQIZSWliqv3bt393WRhOgVVquV\n4cOH88orr2A0GjtME/DMM8/w4osv8vrrr7NlyxYsFguzZ8+msbGxj0osxIk71vWuUqmYPXu2x/3+\n22+/7aPSCtEzP//8M7fffjubNm1i9erVaLVaZs2aRU1NjbLNidzjB3x3tfHjxzNy5EjeeustZVlS\nUhLz5s3jqaee6sOSCdG7lixZwooVKySwEQOe2WzmjTfe4LrrrgPaMnFGRERw5513cv/99wNtmTYt\nFgvPP/88Cxcu7MviCtEjR17v0NaSU1VVxVdffdWHJRPi5LBarfj5+fHvf/+bCy644ITv8QO6Jael\npYXt27czZ84cj+Vz5sxh48aNfVQqIU6e7OxsIiMjSUhI4OqrryYnJ6eviyTESZeTk0NZWZnHvd5g\nMDB16lS514sBSaVSsWHDBkJDQ0lOTmbhwoVUVFT0dbGE6BX19fW4XC4CAgKAE7/HD+ggp7KyEqfT\nSWhoqMdyi8VCaWlpH5VKiJNjwoQJLF26lB9++IF33nmH0tJSJk2aRHV1dV8XTYiTqv1+Lvd6caY4\n99xz+eCDD1i9ejUvvPACmzdv5pxzzqGlpaWviyZEjy1atIhRo0YxceJE4MTv8f06hbQQovvOPfdc\n5eehQ4cyceJE4uPjWbp0KXfffXcflkyIvnPkWAYhBoIrr7xS+TktLY0xY8YQGxvLN998w2WXXdaH\nJROiZxYvXszGjRvZsGFDt+7fXW0zoFtygoOD0Wg0lJWVeSxvn3NHiIHMZDKRlpZGVlZWXxdFiJMq\nLCwMoNN7ffs6IQay8PBwoqKi5H4vTmt33303y5cvZ/Xq1cTFxSnLT/QeP6CDHJ1Ox5gxY1i5cqXH\n8lWrVkmqRTHg2Ww29u3bJwG9GPDi4+MJCwvzuNfbbDY2bNgg93pxRqioqKCoqEju9+K0tWjRIiXA\nSUpK8lh3ovd4zZIlS5acrAL3B76+vjzyyCNERERgNBp54okn2LBhA++99x5+fn59XTwhes0999yD\nwWDA5XJx4MABbr/9drKzs3nrrbfkWhenPavVSkZGBqWlpbz77rsMGzYMPz8/HA4Hfn5+OJ1Onn76\naZKTk3E6nSxevJiysjLefvttdDpdXxdfiOPS1fWu1Wp54IEH8PX1pbW1lZ07d3LTTTfhcrl4/fXX\n5XoXp53bbruN999/n08//ZSoqCgaGxtpbGxEpVKh0+lQqVQndo93nwHefPNNd1xcnFuv17vHjh3r\nXr9+fV8XSYhed9VVV7kjIiLcOp3OHRkZ6Z43b5573759fV0sIXrFmjVr3CqVyq1SqdxqtVr5+YYb\nblC2WbJkiTs8PNxtMBjc06dPd+/du7cPSyzEievqem9ubnbPnTvXbbFY3Dqdzh0bG+u+4YYb3IWF\nhX1dbCFOyJHXefvr0Ucf9djueO/xA36eHCGEEEIIIcSZZUCPyRFCCCGEEEKceSTIEUIIIYQQQgwo\nEuQIIYQQQgghBhQJcoQQQgghhBADigQ5QgghhBBCiAFFghwhhBBCCCHEgCJBjhBCCCGEEGJAkSBH\nCCHECZs+fTozZszo62J0UFRUhNFoZM2aNX1WhjfeeIPY2FhaWlr6rAxCCHGmkiBHCCFElzZu3Mij\njz5KXV1dh3UqlQqVStUHperao48+ysiRI/s0ALvxxhux2+289dZbfVYGIYQ4U0mQI4QQoktdBTmr\nVq1i5cqVfVCqo6uoqGDp0qX86U9/6tNyGAwGrr/+el544QXcbneflkUIIc40EuQIIYS89B4HAAAG\nPUlEQVTols4e1LVaLVqttg9Kc3QffvghAJdddlkflwSuvPJK8vPzWb16dV8XRQghzigS5AghhDiq\nJUuWcO+99wIQHx+PWq1GrVazbt06oOOYnNzcXNRqNc888wxvvvkmCQkJeHt7M2vWLPLz83G5XDz+\n+ONERUVhMpm45JJLqKqq6nDelStXMm3aNMxmM2azmfPOO4/09PRulfnLL79k3Lhx+Pr6eiwvKyvj\npptuIjo6GoPBQFhYGOeffz4ZGRkndO4DBw5w9dVXY7FYMBqNJCUlcffdd3tsM3r0aAIDA/niiy+6\nVXYhhBC9o39VvwkhhOhXLr/8cg4ePMhHH33Eyy+/THBwMAApKSnKNp2Nyfn444+x2+3ceeedVFdX\n8+yzz/L73/+e6dOns379eu6//36ysrJ49dVXWbx4MUuXLlX2XbZsGX/4wx+YM2cOTz/9NDabjbff\nfpspU6awZcsWkpOTj1peh8PBli1bWLhwYYd18+bNY8+ePdxxxx3Ex8dTXl7OunXrOHjwIKmpqcd1\n7r179zJ58mS0Wi0LFy4kISGBnJwcPvnkE1566SWP844ePZpffvnlOH7rQggheswthBBCdOG5555z\nq1Qqd15eXod106ZNc8+YMUN5n5OT41apVO6QkBB3XV2dsvyBBx5wq1Qq97Bhw9ytra3K8vnz57t1\nOp3bZrO53W63u7Gx0R0QEOC+8cYbPc5TU1Pjtlgs7vnz53dZ1qysLLdKpXK/8sorHfZXqVTuF154\n4aj7Hs+5p02b5jabze7c3Nwuy+N2u90LFy506/X6Y24nhBCi90h3NSGEEL3u8ssv9+gudtZZZwFw\n7bXXotFoPJY7HA4KCgqAtkQGtbW1XH311VRWViqv1tZWzj777GOmhG7v+hYQEOCx3Gg0otPpWLNm\nDTU1NZ3u291zV1RUsG7dOhYsWMD/3979hMK3xnEc/0z9MiL/FpMyHCMLpVnQpCnU1CCywEoJJRtj\nYzYoio2MLGalxMK/jWIjJSnJkhIlGyyMpqzIpNRMFHchuuMyuHdcv+b3fu3mOc8536ezOp95nuec\n/Pz8D+9FVlaW7u7udHt7+2FfAEB8sFwNABB3hmFE/c7IyJAk5eXlvdn+HDxOT08lSTU1NW9e9+8B\nKZbHVy9JMJvNGh8fV29vr7Kzs+V0OlVfX6/29nbl5uZ+qfbZ2ZkkyW63f2ksv+OrtgEgURFyAABx\n914Yea/9OQg8PDxIkhYWFmS1Wr9c93nP0FuzNV6vV42NjVpdXdXm5qZGRkbk8/m0trYml8v1n2u/\nJxQKyWw2KzU1NW7XBADERsgBAMT0f85AFBYWSnoKK263+8vnG4ahlJQUBQKBN4/bbDZ5vV55vV5d\nXFyopKREo6Ojcrlcn6793O/o6OhTYwoEAlEvagAAfD/25AAAYnqegbi+vv72WnV1dcrMzJTP59P9\n/f0/jl9dXcU8/9evX3I6ndrb24tqD4fDCofDUW1Wq1UWi+XlI6e1tbUxa19eXkp6CkEul0vz8/M6\nPz+P6vN6mZwkHRwcqLy8POa4AQDxxUwOACCmsrIySdLAwIBaWlqUlJSkqqoqWSwWSW8/2P9baWlp\nmpqaUmtrq0pLS1++QxMMBrWxsSG73a65ubmY12hsbFRfX59ubm5e9vycnJzI7XarublZxcXFMpvN\nWl9f1/Hxsfx+vyQpPT3907UnJiZUWVkph8Ohrq4uFRQUKBgMamlp6WVvjyTt7+8rFAqpqakpbvcI\nAPAxQg4AICaHw6GxsTFNTk6qs7NTj4+P2t7elsVikclk+vRytvf6vW5vbm5WTk6OfD6f/H6/IpGI\nrFarKioq5PF4PqzT2tqq/v5+raysqKOjQ9LTMra2tjZtbW1pcXFRJpNJRUVFmp2dfenzldp2u127\nu7saGhrS9PS0wuGwDMNQQ0ND1FiWl5dlGIaqq6s/dY8AAPFheoznX3AAAPwGPB6PDg8PtbOz82Nj\niEQistlsGhwcVE9Pz4+NAwD+ROzJAQAknOHhYR0eHn74XZ3vNDMzo+TkZHV3d//YGADgT8VMDgAA\nAICEwkwOAAAAgIRCyAEAAACQUAg5AAAAABIKIQcAAABAQiHkAAAAAEgohBwAAAAACYWQAwAAACCh\n/AUvyzpaWLpBYQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a048198>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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kq8l7IrbO62shBAEAAAApIOExQXfffbf+8pe/9Lj9r3/9q+6+++5eqVR/cPqz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\nTIKIiIiIiMiuMAkiIiIiIiK78n9kZkgkjVTMVQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a015c88>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Using SymPy to compute Jacobians"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Depending on your experience with derivatives you may have found the computation of the Jacobian above either fairly straightforward, or quite difficult. Even if you found it easy, a slightly more difficult problem easily leads to very difficult computations.\n",
      "\n",
      "As explained in Appendix A, we can use the SymPy package to compute the Jacobian for us. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "sympy.init_printing(use_latex='mathjax')\n",
      "\n",
      "x_pos, x_vel, x_alt = sympy.symbols('x_pos, x_vel x_alt')\n",
      "\n",
      "H = sympy.Matrix([sympy.sqrt(x_pos**2 + x_alt**2)])\n",
      "\n",
      "state = sympy.Matrix([x_pos, x_vel, x_alt])\n",
      "H.jacobian(state)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{x_{pos}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}} & 0 & \\frac{x_{alt}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "\u23a1       x_pos                    x_alt        \u23a4\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  0  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2   _________________        _________________\u23a5\n",
        "\u23a2  \u2571      2        2        \u2571      2        2 \u23a5\n",
        "\u23a3\u2572\u2571  x_alt  + x_pos       \u2572\u2571  x_alt  + x_pos  \u23a6"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This result is the same as the result we computed above, and at much less effort on our part!"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Designing Q"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**author's note: ignore this, it  to be revised - noise in position and altitude is independent, not dependent**\n",
      "\n",
      "Now we need to design the process noise matrix $\\mathbf{Q}$. From the previous section we have the system equation\n",
      "\n",
      "$$\\dot{\\mathbf{x}} = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0& 0& 0 \\\\ 0&0&0\\end{bmatrix}\n",
      "\\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix} + \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "where our process noise is\n",
      "\n",
      "$$w = \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}$$\n",
      "\n",
      "We know from the Kalman filter math chapter that \n",
      "\n",
      "$$\\mathbf{Q} = E(ww^T)$$\n",
      "\n",
      "where $E(\\bullet)$ is the expected value. We compute the expected value as\n",
      "\n",
      "$$\\mathbf{Q} = \\int_0^{dt} \\Phi(t)\\mathbf{Q}\\Phi^T(t) dt$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rather than do this by hand, let's use sympy."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "sympy.init_printing(use_latex='mathjax')\n",
      "w_vel, w_alt, dt = sympy.symbols('w_vel w_alt \\Delta{t}')\n",
      "w = sympy.Matrix([[0, w_vel, w_alt]]).T\n",
      "phi = sympy.Matrix([[1, dt, 0], [0, 1, 0], [0,0,1]])\n",
      "\n",
      "q = w*w.T\n",
      "\n",
      "sympy.integrate(phi*q*phi.T, (dt, 0, dt))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{\\Delta{t}^{3} w_{vel}^{2}}{3} & \\frac{\\Delta{t}^{2} w_{vel}^{2}}{2} & \\frac{w_{alt} w_{vel}}{2} \\Delta{t}^{2}\\\\\\frac{\\Delta{t}^{2} w_{vel}^{2}}{2} & \\Delta{t} w_{vel}^{2} & \\Delta{t} w_{alt} w_{vel}\\\\\\frac{w_{alt} w_{vel}}{2} \\Delta{t}^{2} & \\Delta{t} w_{alt} w_{vel} & \\Delta{t} w_{alt}^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "\u23a1           3      2                2      2             2            \u23a4\n",
        "\u23a2  \\Delta{t} \u22c5w_vel        \\Delta{t} \u22c5w_vel     \\Delta{t} \u22c5w_alt\u22c5w_vel\u23a5\n",
        "\u23a2  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2          3                       2                      2           \u23a5\n",
        "\u23a2                                                                     \u23a5\n",
        "\u23a2           2      2                                                  \u23a5\n",
        "\u23a2  \\Delta{t} \u22c5w_vel                       2                           \u23a5\n",
        "\u23a2  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       \\Delta{t}\u22c5w_vel      \\Delta{t}\u22c5w_alt\u22c5w_vel \u23a5\n",
        "\u23a2          2                                                          \u23a5\n",
        "\u23a2                                                                     \u23a5\n",
        "\u23a2         2                                                           \u23a5\n",
        "\u23a2\\Delta{t} \u22c5w_alt\u22c5w_vel                                           2   \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \\Delta{t}\u22c5w_alt\u22c5w_vel     \\Delta{t}\u22c5w_alt    \u23a5\n",
        "\u23a3          2                                                          \u23a6"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example: A falling Ball"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the **Designing Kalman Filters** chapter I first considered tracking a ball in a vacuum, and then in the atmosphere. The Kalman filter performed very well for vacuum, but diverged from the ball's path in the atmosphere. Let us look at the output; to avoid littering this chapter with code from that chapter I have placed it all in the file `ekf_internal.py'."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import ekf_internal\n",
      "ekf_internal.plot_ball()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+MM4+VRhqtXE62latjK/WrY3vn+QuRBoNNGpkfA0fbkxLTjYOyD98+P7ryBHj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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f914a703f28>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can artifically force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
      "\n",
      "Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
      "\n",
      "$$a_x = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_x \\\\\n",
      "a_y = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_y- g\n",
      "$$\n",
      "\n",
      "These equations will be very unpleasant to work with while we develop this subject, so for now I will retreat to a simpler one dimensional problem using this simplified equation for acceleration that does not take the nonlinearity of the drag coefficient into account:\n",
      "\n",
      "\n",
      "$$\\ddot{x} = \\frac{0.0034ge^{-x/20000}\\dot{x}^2}{2\\beta} - g$$\n",
      "\n",
      "Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
     ]
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