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"source": [
"[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"# Nonlinear Filtering"
]
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"\n",
"\n"
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""
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"source": [
"#format the book\n",
"%matplotlib inline\n",
"%load_ext autoreload\n",
"%autoreload 2 \n",
"from __future__ import division, print_function\n",
"import matplotlib.pyplot as plt\n",
"import book_format\n",
"book_format.load_style()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Introduction"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"The Kalman filter that we have developed to this point is extremely good, but it is also limited. All of the equations are linear, and so the filter can only handle linear problems. But the world is nonlinear, and so the classic filter that we have been studying to this point can have very limited utility.For example, the state transition function is $\\mathbf{x}^- = \\mathbf{Fx}$. Suppose we wanted to track the motion of a weight on a spring. We might be tracking an automobile's active suspension, for example. The equation for the motion with $m$ being the mass, $k$ the spring constant, and $c$ the damping force is \n",
"\n",
"$$m\\frac{d^2x}{dt^2} + c\\frac{dx}{dt} +kx = 0$$\n",
"\n",
"which is a second order differential equation. It is not possible to write a linear equation for this equation, and therefore we cannot even formulate a design for the Kalman filter. \n",
"\n",
"The early adopters of Kalman filters needed to apply the filters to nonlinear problems and this fact was not lost on them. For example, 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",
"\n",
"$$x=\\sqrt{slant^2 - altitude^2}$$\n",
"\n",
"And then, of course, the behavior of the objects being tracked is also nonlinear. So shortly after the idea of the Kalman filter was published by Kalman people began working on how to extend the Kalman filter into nonlinear problems. \n",
"\n",
"It is almost true to state that the only equation we (anyone) know how to solve is $\\mathbf{Ax}=\\mathbf{b}$. We only really know how to do linear algebra. I can give you a linear equation and you can solve it. If I told you your job depended on knowing how to solve the equation $ax+b3=c$ you would trivially solve it for any values of $a$, $b$, and $c$. There is no linear equation that you cannot either solve or prove has no solution ($x+1=x$).\n",
"\n",
"Anyone with formal mathematical education has spent years learning various analytic ways to solve integrals, differential equations and so on. It was all a lie, nearly. Even trivial physical systems produce equations that no one in the world know how to solve analytically. I can give you an equation that you are able to integrate, multiply or add in an $\\log$ term, and render it insolvable. It is stuff like this that leads to jokes about physicists stating stating the problem \"assume a spherical cow on a frictionless surface in a vacuum...\"\n",
"\n",
"How do we do things like model airflow over an aircraft in a computer, or predict weather, or track missiles with a Kalman filter? We retreat to what we know: $\\mathbf{Ax}=\\mathbf{b}$. We find some way to linearize the problem, turning it into a set of linear equations, and then use linear algebra software packages to grind out a solution. It's an approximation, so the answers are approximate. Linearizing a nonlinear problem gives us inexact answers, and in a recursive algorithm like a Kalman filter or weather tracking system these small errors can sometimes reinforce each other at each step, quickly causing the algorithm to spit out nonsense. \n",
"\n",
"What we are about to embark upon is a difficult problem. There is not one obvious, correct, mathematically optimal solution anymore. We will be using approximations, we will be introducing errors into our computations, and we will forever be battling filters that *diverge*, that is, filters whose numerical errors overwhelm the solution. \n",
"\n",
"In the remainder of this short chapter I will illustrate the specific problems the nonlinear Kalman filter faces. You can only design a filter after understanding the particular problems the nonlinearity in your problem causes. Subsequent chapters will then design and implement different kinds of nonlinear filters."
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"## The Problem with Nonlinearity"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"As asserted in the introduction the only math you really know how to do is linear math. Equations of the form \n",
"\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. 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",
"\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 perfect 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 multiply 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 I 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",
">\n",
"$$L(f(t)) = f(t) + 1$$\n",
">\n",
">is not linear because $L(0) = 1$! Be careful!"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"## An Intuitive Look at the Problem"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"I particularly like the following way of looking at the problem, which I am borrowing from Dan Simon's *Optimal State Estimation* [[1]](#[1]). Consider a tracking problem where we get the range and bearing to a target, and we want to track its position. Reported distance is 50 km, and the reported angle is 90$^\\circ$. Now, given that sensors are imperfect, assume that the errors in both range and angle are distributed in a Gaussian manner. Given an infinite number of measurements what is the expected value of the position?\n",
"\n",
"I have been recommending using intuition to solve problems in this book, so let's see how it fares for this problem (hint: nonlinear problems are *not* intuitive). We might reason that since the mean of the range will be 50 km, and the mean of the angle will be 90$^\\circ$, that clearly the answer will be x=0 km, y=90 km.\n",
"\n",
"Well, let's plot that and find out. Here are 3000 points plotted with a normal distribution of the distance of 0.4 km, and the angle having a normal distribution of 0.35 radians. We compute the average of the all of the positions, and display it as a star. Our intuition is displayed with a large circle."
]
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cduXKlR16Tl5eXjh8+LC8imY2m/Hggw9i6tSpOHTokNxu2bJluOGGG7B69eoGcyzGjx+P\nbdu2ybEPHToUTz31FP76179i1apVAID58+cjICAAb7/9tgy29+7di48//hiHDx/GlClTrPa3cOFC\npKWlISYmRt5+5swZnD59Wv6uo6OjER4ejr179yIhIQFjx47FDTfcgL1792L27Nm49tpr5WNFsiAt\nLQ033nijvH3dunUd+v11VIvB9vnz57FgwQJcunQJgwYNwqRJk3Ds2DEMGTIEALBixQpUVlYiISEB\nJSUlmDhxIg4dOtRr6oGIqOeyzGomJyfLQDAhIQEZGRnIz89HVlYWVCoVDAYDlEolfHx8oNfrkZOT\ng4CAABgMBpSXl6N///4y8M3JyUFxcTE8PT3lpD8xITA4OBiFhYXIzc1FSEgIQkJC5H1OTk4yqy2C\n1EGDBgH4NVC0zB5HREQgPj5eZmxFdllMLBQBsKenJ1xcXDrckcTV1RWBgYFYvXo1kpOTG0zk1Ol0\nUKlUUKlUCA0NRWFhIUJCQuTkTdE5RZwElJSUoLa2VrYrFOU0lvu9cuUK6urqMGjQIAQEBKCgoAAG\ngwEGg0Feui4pKQFQH3BrNBoYDAYYjUao1Wo5KbOgoAAajUZ2QsnPz5cTTQsLC5GTkyNLdGyvBHCC\nJbXVXXfdhUcffRQffvghZs+ejQ8//BAbNmxosN2+ffvQr18/TJ8+HZcuXZK3jxw5Er6+vkhPT5fB\ntgi0TSYTysrKUFNTg8jISJjNZnz55ZcIDAyEh4cHXF1dkZ6ejvj4eHni2lEPPPCADLSB+hI6vV6P\nlStXWo0bAG699Va8/vrrqKqqssoEL1myRH6tVCoRERGB8+fP4/7775e3e3t7Y+TIkbLkTPyOQkND\nMXr0aKufNXXqVCgUCqSnp1sF29HR0VYnNWPHjkX//v2t9tkU8R7w73//G+PGjbOque9OLY7C9nJJ\nY9auXYu1a9d2yoCIqO9qTY9mrVaLxMREaDQaWZto2VIvJydHtrNTq9UykBMfdCLozcnJQUFBgcxG\ni2yy6Bgi6PV6GQAGBARg9erVMtMq+Pj4ICoqCikpKTLQFB1KRKAqAmdLSUlJ0Gq1iI+PlyURlqUS\noi2e0WjEkiVLsGXLlg79fu+//37ExcXBzc3NqluIZSs+Mf6kpCTZS1sEuFFRUYiMjJTPXwQClllw\ntVots9MBAQFwcnKSNewiW5aRkWHVuUUE0+L3L7LuWVlZyMzMhFqthtFoxGeffSZPqCw7qWg0GuTn\n56OkpAQajQZJSUnyw1ucvFjWsQKtW2iIup6Y89DdCTu1Wo3bbrsNu3fvhlKpRGVlJe65554G2+n1\nely9erVBVzZBXC0DgJMnT2LFihXIyMhoUKssyhTc3NywceNG/PnPf4afnx9uvvlmzJgxA/feey8C\nAwPb/XyGDx/eYNwArAJlSwqFApcvX8bgwYPlbZYZZKA+sHZxcWmwDkv//v2tnrder8e3334rEw+2\nP8dy28Z+DmD9Xt6cqKgozJs3D88++yw2bdqEqKgozJw5E3FxcfI9vjv0jJCfiAi/toazDIBEcCbu\ns+xPLSbJiGyGaL8nJhxVVFTIrydPniwD8oyMDPkG379/fxiNRlRUVMjsaEVFBdRqNV544QUZyIeF\nhcmALSoqSk4QF/uMjIy0CqYt+0fbBnOWl2fFcxXbWAaFYns3NzcsW7YM7733XrtXkRwwYACWLVsm\n60ctF6MRP1sE/4LlNpbPRaPRyMDb8j7L7JTBYIBGo5GlM5ZZZ/E48bvNz89HbW0tBg0ahOLiYpSU\nlCA4OFhekaisrISLiwuqq6tx/vx5qNVqLFiwACkpKcjJycGsWbNkEF5YWIiEhASrqxGAdZ9vITIy\nUrZl7My6eGq/FStW4PLly9izZ093DwVxcXFYtGgRrly5gpiYGFkbbMlkMmHAgAF47733Gt2HOCEt\nLS1FdHQ0vLy8kJiYiBEjRsDDwwM//fQTFi9ebDXJevny5Zg1axZSUlKQlpaG559/HomJifjPf/7T\noI7almVpliXL8hUxbqB+FXBxRc+W7fNtrAtLU53nLLuEmEwmjBkzBn//+98b3TYgIKDFn2O7z+bs\n27cPx48fx3/+8x+kpaVh6dKl2LBhA44dO9ZowN8VGGwTUZezzSqKYCcyMhI6nU7eJhZRAWBVxiCy\nxWfOnAHwa1kDAJm9KC8vl+UhAQEByMzMRGVlpSxfECIiIqwyqyKQFOMTAZpl9jkyMrJBYGq5QEtT\nvaK1Wm2z2dPm7hs7diySkpKwaNGiNpeTuLi4YNeuXRg7dmyL27aU3W3ufrGwjU6na3DSYHnyJIjj\nKU4AxImO0WiUmW8XFxfU1dUhJCQEubm5qKurQ0BAgCzFmTx5styf+DvYu3ev1ZUKcSIlMuaWX4us\nN7PaPcMrr7xiFXh2p1mzZsHNzQ2ZmZnYuXNno9sMHz4cn3zyCW6++eZms/Hp6em4fPky/vWvf1nV\nLYvJ2raGDh2K5cuXY/ny5Th//jyuv/56/PWvf5XvTWq12upkEqjvXCReUy0Rme5+/fph2rRprXpM\ne40YMQJZWVmd+nNa6nM+YcIETJgwAc8++yxSU1MxY8YMbNu2TS7Q1dUYbBNRt7IMqpOSkqDT6WQp\nhfgwCQsLk0GTSqWS5W2iPEEEZgAaBNKijKCgoEBmfURWuqCgAIWFhTIDKz6oLDO64iRABGOWQVlT\nAVpLgZvtEuetCfQUCgVcXFywc+dOPPLII63OcA8YMAC7du1CTExMkx9Q7Q0027riZlPEiZTIlou/\nidzcXACQrWQts1IVFRUySM/IyEBJSYmc7CmucAD1J101NTWyDl2v18vg25blJFvbrD11DdHppyfw\n8PDAli1bkJeXh9mzZze6zfz587FlyxY899xzsn2oUFdXh7KyMvj4+MjnZHkiYTKZsGnTJqvHiPIS\ny0z04MGDMWjQIKuOGMOHD7dKRgDAm2++2eoTlfHjx2PEiBHYtGkT7r33XvTr18/q/uLi4lZlgVuz\nuM8999yDgwcPYsuWLXjooYes7jMajaipqWnw81siTmx+/vlnq7ITg8EAb29vq3HdcMMNADreUaQj\nGGwTUadrrga2sfsMBgMyMjIQExMjg1/g10ylqKOura1FeXk56urq4Obm1uCN3rKcRJR96HQ6FBYW\nykulhYWF8mdYZjkss+yWGejWBlsdyQi31ty5c2E2mzFmzBhs3boVb731VpNZbldXV9x///1YtmwZ\nxo4d22kr3rVU3xwXF9fq7L34nYt6cNuTEDEJs6KiQq54KY6j+BsRX1tm+UQXlCtXrsgyIfG3IVbs\nFCdnlhNExSTb0NBQrFixQtaENzeHgHq3hQsXNnq7KGmYMmUKEhIS8OKLL+Lrr7/G9OnT4ebmhrNn\nz+Kf//wnnn/+eSxatAi33HILBgwYgD/+8Y949NFH4ezsjP3798tuRsK3336LadOm4e6778bo0aPh\n5uaGgwcP4syZM3j55ZfldkuWLMGyZcswb9483HrrrcjOzsahQ4cwcODAVpVbKBQKbN++HbGxsRg9\nejTuu+8+DB48GAUFBTKI//TTT1vcT1M/y/L2hQsXYv/+/XLiupgU+u233+L999/H/v37ZYen1v6c\nCRMmAABWrVqFBQsWwNXVFb/97W+xZ88eJCUl4c4778SwYcNQWVmJ5ORkODs7Y968eS0+H3thsE1E\nnc6y9tqSmNwothFdLIxGI4qLi5GbmwtnZ2dZJiICKDExRrSPA+ovo06ePBmhoaHIyMiQNcLAr5Pi\nkpOTER8fb5UttWwTZzs+RwiqFAoFxo0bh1deeQVLlizBmTNnkJWVJRd6CQ8PR0REBEaNGoUxY8bI\n7G9Xa+3vr6ngXJTqJCQkICUlBSqVCrNmzZLHUhDH9IUXXpDHOCMjw2qhEDc3NxgMBuzYsQO1tbVw\ndnZGSUmJzG5HRkYiISFBTqwV2W+gPgAXJUKixjwqKqrVJ5LkWFpzUmrby3rz5s248cYbsXXrVqxZ\nswbOzs4ICgrCPffcI0sn1Go1PvroIzz55JNYu3YtvLy8MHfuXCxbtgzjxo2T+7r22muxcOFC/Pe/\n/4VWq4VCocDIkSNlH2/hgQceQH5+PrZv347U1FRMnToVaWlp+O1vf9vgOTT1nKZMmYJjx47h+eef\nxxtvvIErV65Ao9FgwoQJVp1Hmurd3drbFQoF/vWvf+HVV1/Fzp07kZKSAg8PDwwfPly28muJ7c+J\niIjAhg0b8MYbb+C+++6D2WxGeno6fvOb3+DEiRPYt28fLly4gP79++PGG29EUlKSDNC7g8LcmWtd\nNsMyfe8I69hT0xxhwQVqnc4+lk0tMCJuF5lDHx8faDQaZGZmora2Vl4SFO3yFi9ejJSUFAD1dZN7\n9+6F0WiUvZ7VarXVhEXbS/+WpSkiY2pZG96aYMhRAqeq/HxAqcTZsjIAwOjRo5v8EOxKnfn7a27h\nItGZRtSLW7aAFJ1ORJ2+OJkLDg5GQUGBPHET9eDV1dVwdXWVSzOL8iQR5CcmJsq+4rZZ+M58zo7w\nHmvbFo7I0TX3N93RGJaZbSJqk7YEFKIsBIAMeCz7IqvVatkeLzIyUl66FwuhiEuLYra6aL8nMt8i\nw9lYja1td4mWyhts9fQgW/qlrrnqmmsAwKqXbnfqzN9fc3Xyq1evtrpNXLWwbF+YkZGBqKgo+Xcj\nepunp6cDqF+Gvrq6GgqFAs7OzrJ/d25uLlQqlfxbioqKQkZGBvR6PRISElo8cXOUEzYisi8G20TU\naURQIVZoFJflRccI0QZOTGQTy3iLAAaAvPzv6ekpAydRHmKZpQYaBtS24+jtTCYTFJ9/DigUcJoz\nR04Y7SsaKwNq7NiLxXAs70tOToa/v79Vn2+VSoWKigr4+PjIXuxhYWFWfdxFaZP43vLqjbhyYnuF\nh4j6NgbbRNQiywydbRs725Zuli36AFgt3Q1ArtwoJrEZjUarriNixUexBLhlS0AxlrZmqXurmh9/\nhMv27YBCAfWMGbjUQ1ZL60lsu8kIYjVP8bXtIj6WK2yKOQGWGpvUa9kesi0lS0TUu/GdmYjaxbJF\nHwBkZmbKmmrRNSI/Px8VFRXIyspCVlYWIiIiZH9jADK7DfyatRZZccustWWQ31dVFxXBbNO5wJSf\nD+UvJzZexcVQDBgAY16evF/h6QnXJla262ts/3Ys6/st77edCAn8ukKlWHgHgOyQIm4H6hdkiY+P\nx9SpUxEaGgqdTsdgm4gYbBNRQ5YZPkEEJSIQERPMPD09oVKpoFarZe9jsVAMUL/IjOh7bLmqn5jQ\nJrqTiJ9lu1iMpb4cuCicnVH3v//BbflyKETQXVUl77/mrrtwzS+Te8yenjC+9hqcf/Obbhhpz2JZ\n2gSgQW2/ZRmIKGcKDQ1FYWGhrPUODQ2V9d2iHaVKpbJa+U5cqamrq0NWVpYsQWlNbTcR9W4Mtomo\nUaKGWgQner2+QYlIXV2d7FkslhcG6gPpwMBAuLm5yf7GAQEBVovHsBykbVwGDIDTnXfCOHgwXBMS\n4PTll1b3K4xGwGhE3Q03oPqNN+B20009ZrJkT9DSIjWWq4UmJSXJ8hBRwiRWIl29erU8QRR92hMT\nE5GRkSF7gFuuWpmVlQW9Xt+gM49tZxUi6r0YbBNRA5at8oBfA5GUlBTZPQSADK7DwsKg1+tlYB0T\nEwMfHx+ZBbTsONKachAGH41TKpVwnzgR1fv3o+7FF+G6davV/eX33w/np5+G+9Ch3d76r6dpzd+U\nZTs/0VZSPE708xbf63Q6GXRblkLp9Xo596C4uFi2tbSd26DT6WRnFMv7ASAvLw+xsbHtf7JE1KMw\n2Cbqw1JTUwH82s/XcmKXoNPpZGAggmlxqd1yWWyxWiPwa2AiWqwBaHLJc2obhUIBl6FDUTN8eIP7\n6oYPh4qBdqewbTdouZCOuCITExODnJwcqFQqhIWFWb1uPvvsM9TV1SEgIACFhYWyF7x47aSkpMjF\nmsTrC6ifrHnx4kVkZ2djzpw5fK0Q9QIMton6GMsMWnZ2NsLDw+XtIqBITExEQUGBrL3Oz8+HWq3G\nrFmzoNfrkZOTI8tJAgICEBoa2mhQYJkFpM5Te/EinN98s/7r224DzGY4HzoE1Y4dqImPh6u/fzeP\nsPdpbBFxM/WVAAAgAElEQVSbtLQ0ucJlYWGhPFHV6XRwdnaGp6enzHaLkhJLtbW1Vq8l0eLy3Llz\nDX4WS06IHBeDbaI+SGTSSktLER4ebrXwB1C/yAcAGWgLIvsGwKrVmdBUwM0AoXOZcnPhkp+Pqldf\nhfKuuwAAV3bvhtfTT6M2NxdgsN1lRBAuViwVWfDg4GD5egKAkpISVFZWwsXFRa6iCkB258nJyZHB\n9qVLlzBw4MCufBpEZEcMton6ANs+2TqdDoWFhVCpVADqA4WCggIZMAQHB8sPflFzLb4PDQ21CrIb\n6ynM4Np+zGYzTIWFqEpLg2tkJJxcXAAAObfeisFjxkB14QLMZjNLSbqAeF3ZdtCx7O0tJlqq1Wq4\nubnJ7jwGg0FOohQrqwL1J7Qi0E5MTLQq6+LrisgxMdgm6iNsSzkMBgOKioqQnZ0t2/gZDAbMmjXL\naoKkWPjDtu81dY/aykoobrwR7sOHWwXUNSYTzvn7Y3T//qitrITLLydS1DWausIjFs+xnQeRkpIi\nT3YNBoPsbLJixQoYjUYEBwdbbW+7XyJyHG0Ktjds2ICnn34aCQkJ2Lx5MwBg8eLFeOedd6y2mzhx\nolymmYi6VmO1nSKbLXpki4U4TCYTUlNT4eLiAmdnZ1RUVMhSEfF4MbEL+LX8RKfTWWXzGqtnJftw\nUangMmJEo/eZzWZ4hIR08Yj6rtYEv01lpS0nEVsG08nJyfjqq6+Qn58PNzc3uS3Q8LUteofz9UfU\ns7W6CeuxY8ewbds2jBs3ziqbolAoEBMTgwsXLsh/Bw8etMtgiah5lpMcxZLqCQkJcmENAHJVx5qa\nGgCAu7s7IiIiMHnyZKtFOoD6D3XRQYGTHInarrE5C+L7nJwcANZXjQCgymKxIst5Epa0Wq3sfd+b\nVlY1mUww/bIiZ1fbsWMHlEollEoljhw50ug2I0aMgFKpRHR0dBePjhxZqzLbpaWlWLhwIZKTk7Fu\n3Tqr+8xmM1xdXeHr62uP8RFRK9m27RM9gMUiMiI75ubmhtraWjg7O8PPzw/33Xcfhg0b1uR+LTPY\nTQUNRNQ2Yg6EOImNjIyEVquVAfhvfvMb+Pr6Qq/Xy9ewuMokrjBZXnESHP01WfP99zAbjXD/pYa9\nO3h4eECr1eKWW26xuv3YsWPIy8uDu7s750RQm7Qq2F66dCnuuusuREVFwWw2W92nUChw5MgR+Pn5\nyYUr/vrXv2LQoEF2GTARNU6n0yEjI0MG1QUFBVCpVFi9erW8T6PRyFZkQP0iHLGxsVZ9thvj6B/g\nRD2NWMxGsFzsyd3dHeHh4cjIyEBmZibc3Nysrjrt3bsXAGSwDcDqRLuxScuOwpybC1RVwTxqVLcF\ntL/73e/w/vvv47XXXoOz869hklarxahRo+Dk5NQt4yLH1WIZybZt25CXl4f169cDQIM//tjYWOza\ntQuffvopXn75ZXzxxReYNm0aqqur7TNiIpIlIlqtFjExMbJ2Mz8/H7m5uTLQnjVrFuLi4hAZGWnV\nhiw+Pt7qg1qUmohL3mLfRGQf4rVmWWYSFxeHsLAweHl5WW2rUqkQFRWF+Ph4JCUlQaVSQaVSITIy\nEnq93qrtoGUpmU6nc6jXsclkguLzz6H8z3+6rZQEABYsWICff/4ZH3/8sbytrq4O+/btwx/+8IcG\n25vNZmzevBljx46Fh4cH/Pz8sGTJEly+fNlquwMHDuD3v/89hgwZAnd3dwwdOlROiLW0ePFieHh4\noKCgALNnz4aXlxd8fX3xl7/8BSaTyT5Pmuyq2cz2t99+i6effhpHjhyRZ3Jms9kqu33PPffIr8eM\nGYOIiAgEBQXho48+wpw5cxrd74kTJzpj7NTNeBy7zsaNGwEA4eHhyM7Otvr6u+++Q2lpKYKCguDn\n54eCggLU1NRg0qRJ0Gg0uOuuu3Du3Dnk5+ejrKwMX331FS5evIjw8HC5JPTx48cB/HpM8/LyrL4n\nx8Fj5tg2bNhg9b14bWs0Gpn4MhqN8PLykqu4fvfddzh+/LhMil28eFE+Pi8vDydOnJCrxcbGxlp9\n3V5BQUFwd3dv9+MbU/PDD3B5+20orl6F8ckn4TRqVKfuv7UCAwMxZcoUaLVa3H777QCATz75BBcv\nXsSCBQvklQXhoYcewttvv43Fixfjscceww8//IDNmzfjiy++wPHjx+VE1x07dsDDwwPLly+Ht7c3\njh49ildeeQU//vhjg32aTCbExsbi5ptvxssvv4y0tDS8/PLLGD58OJYtW9Y1v4g+pqysDCdPnmz0\nvpAOTjxvNtg+evQoLl26hDFjxsjb6urq8Pnnn+Mf//gHysvL4fJLj1dBo9EgMDAQZ8+e7dDAiKhp\nItA+fPgwgoODcenSJeTk5CA2NhZlZWUAgJUrV8oP1UuXLgGoX+1x4MCBVoE2AMycOdNq/x35ECai\nzmH5OhRfi9e0sHLlSmzcuBGHDx/GgQMH8MYbbyA7OxtZWVmIiIiQwXV2djbOnTtntWqs5f66+jVf\n9dNPwC/vS4L5xx+h/GU1TXN2ttVEUQDAgAFwHzLE7mNTKBSIi4vDE088gcrKSnh4eGDPnj2YOHFi\ng/ktmZmZePPNN7Fr1y6rrHdsbCymTJmCd955Bw888AAAYM+ePfDw8JDbPPDAAwgJCcGaNWvw4osv\nIjAwUN5XU1ODu+++G2vWrAFQX84bERGB7du3M9h2QM0G23PmzMFNN90kvzebzfLy8+rVqxsE2gBQ\nXFyM8+fPy8kfjRH1oeSYROaMx9H+xCVgcZVITHqcM2cO3n33XVRUVKCiogLDhw+HwWCAwWDA9ddf\nD6D++Oj1egwbNkxOYLZtESaOpXhDJ8fF12XvYXksbY/n+PHj5fLu4j6DwQA3NzdMmDAB48ePh8Fg\nQHV1NQwGA5KTkwHUv2eI8pI1a9bIbiYieGzr302DQLiNXAYNQvWZM3BbvBjK8+cb3O8xf7782jRw\nIIw7dsClC9ta3nXXXXj00Ufx4YcfYvbs2fjwww8bXHUAgH379qFfv36YPn26TGoAwMiRI+Hr64v0\n9HQZbItA22QyoaysDDU1NYiMjITZbMaXX35pFWwDkI8TbrnlFuzevbuznyr9wsvLq8nXQWlpaYf2\n3Wyw7e3tDW9vb6vbVCoV1Go1Ro8ejatXr2LdunWYN28e/P398f3332PVqlXw8/NrsoSEiNrGtvOA\nuA2AXBLadgU728cDv3Y7ADjhkcjRiZZ/oj1naGio1cm0j48PCgsLUVhYiNWrVyMxMREGgwFhYWFy\njgfQfT26ndzc4P7b38L48cdwSkyESxPvX7V33IGa9evhbtN22N7UajVuu+027N69G0qlEpWVlVZl\ns4Jer8fVq1fh5+fX6H6Ki4vl1ydPnsSKFSuQkZGByspKq+1sgzlXV9cG+1Sr1SgpKWnvU6Ju1OYV\nJBUKhfyDd3Z2xsmTJ7Fr1y4YDAZoNBpMmzYN+/fvh6enZ6cPlqgvsG3hp9frZSZLo9HIoNvya/Gh\nK74WxG3skU3k2CxPlG37ctt+LzqdiBN1oaKiAoWFhfJ9wzJg7w4KhQLuY8ag5tVXYRw9Gm42V9iq\nV6wAHn8cHs1cKbenuLg4LFq0CFeuXEFMTAwGDhzYYBuTyYQBAwbgvffea3QfarUaQH0wHR0dDS8v\nLyQmJmLEiBHw8PDATz/9hMWLFzeY+MjWgr1Lm4Pt9PR0+bW7u3uD+jEi6jixWAUAFBYWoqCgQHYY\nAeo/TEUXAo1GI7PWlllsobFFNYjIcTX1mm5qRUmdTie7EYn3FpENt10KvjuufjkNGACTq2vDO0wm\nODeRMe4Ks2bNgpubGzIzM7Fz585Gtxk+fDg++eQT3Hzzzc0mGdPT03H58mX861//wpQpU+TtaWlp\nnT5u6nnaHGwTkf1ZZp4s5z+ID0yx7LrY1rLUxBF76xJR85p7TdtewRIBs+UVLb1eLzPe4kQ+MjIS\nOp1O/hO3dbXaoiI4/+MfAIDqJUsALy+4vPIKnHfuRM3DD8MtOLjLxwTU11hv2bIFeXl5mD17dqPb\nzJ8/H1u2bMFzzz0nu0YJdXV1KCsrg4+Pj+zoZpnBNplM2LRpU6P7ZWa7d2GwTdQNLEs9LBefSEhI\nsKrPLiwshEajwerVqwH82kdX9NWNioqSH5gAGGgT9SG2WWiR0Ra3i8BZvD+ICdaWPfdF4B0aGgq9\nXt8t7yGms2fhcuECqvbsgVNsLBQuLqiaMgVu991Xv8hNNwXbALBw4cJGbxctkKdMmYKEhAS8+OKL\n+PrrrzF9+nS4ubnh7Nmz+Oc//4nnn38eixYtwi233IIBAwbgj3/8Ix599FE4Oztj//79KC8vb3b/\n1Dsw2CbqRiIjbZlNEtlpnU4ny0fEB6FYrll8WIoPRsvV54iob7MNlnU6HbKysnDlyhX0799fntDH\nxcVZ1XVbTsLuKmazGXXl5aj69FO4jR8PpbJ+rT2n2bNRFRQE848/wmQyydvtrTUZZcu5awCwefNm\n3Hjjjdi6dSvWrFkDZ2dnBAUF4Z577sG0adMA1Nduf/TRR3jyySexdu1aeHl5Ye7cuVi2bBnGjRvX\n7P5bup16PoW5i06fLGfa2nY4IcfCFmPt01gtpMhkA/VZbNFZRNReismRtkuux8fHd0r2icey9+Cx\n7D06+1hqtVokJiaioKAAERERiI+PlyfukZGRsj2g0JpytKqqqk5Z1KbWaERdURFchwxpEEiazWbU\nnD8PhY8PXPr16/DPImpOc3/THY1hmdkm6kStnVxkG0wD9b1yxT5EUG05ecYye81SESJqLdurXyKj\nDfw6/0Oj0cj/LSdoW77X2OP9x9nNDc7XXtvofQqFAq42vaeJHBGDbaJOZjm73/bDSUxEsuwGYDkZ\nUmxj2dbPEoNsImovy/cP0ZvfchK2eN8RSQCdTtdoSQoA3HnnnV03cCIHx2CbqBM1FQyLtnyWtZGW\nl2ljYmJk/9vIyMhu6QhARH2D5fwO0aEEqA+kU1JS4OPjA+DXBIAod4uPj+d7E1E7MNgm6mSW/Wob\nW0xGZKxF1kh88Il6bSKirtBYiYigaWIhGfGYji7XTtSXMNgmsiORGRLZo4yMDOTk5CAsLMwqo9RZ\nEx6JiFrS1HuNZdZadERKSEiwaiHI9ymitmOwTdROrZ0sJFpsAZAdRdq6DyIiexLvQY3VaFu+b4nl\n3XNzczF27NguHyeRI2KwTdRJYmJiAFgvvxsaGoqMjAzExMTITLblghJERD2NZUmJZbZbr9cjMTER\nOp0OixYt6o6hETkkBttEHSTa+AH1vbLF95GRkUhKSpJBuCWu9EhEjkK0DkxISEBOTg5SUlIwb968\n7h4WkcNgsE3UCURfbAByNr9tvbb4wCIi6smaq+nW6/XIysqCyWTq4lEROa6uWf+UqBcSwbO4zJqT\nkwMACAsLk6UijXUjISJyNKK0JD4+HiqVCq6urt08IiLHwcw2UTvYTmyMioqSC9WI5dZF32xms4mo\ntxCrT4pe3B1lNptx+fJlnD17FgUFBSgpKQFQvyR2QEAAhg8fDl9f3wZLuRM5EgbbRG0kFngIDQ1F\ncnIyADS62qMoKxGz94mIHJXt6pM//PBDh/d58eJFHDlyBC+++CKOHTvW6DbXX389Vq5ciaioKPj7\n+/fqoHvo0KGIjo6WnyvNWbduHZ577jmW8zgIlpEQtYMIrgsLC61u1+v10Gq1SEpKQlpaGldbIyKy\nYTab8c0332DZsmWYO3duk4E2AHz11VdYsGABFi1ahK+++gpms9lu49qxYweUSiW++OKLNj2usrIS\n69ats2rr2h4KhcLqZKKgoADr1q1DdnZ2i9tSz8Zgm6iNxCqPer0eBQUFyMrKQmRkZKPZbSKi3sjX\n17ddjzObzfj6669x55134oMPPmj14z755BPccccdyMrKsmvA3R7l5eV47rnnOhxs6/V6bNu2TX5f\nUFCA5557rtFge82aNaisrOzQz6Ou06Zge8OGDVAqlXj00Uetbl+3bh0GDx4MlUqF6OhonD59ulMH\nSdRVtFptg2WLm7ovNDQUKpUKQP1ESFFawpIRIqLGnT9/Ho888gjOnj3b5scWFBRg8eLFyM/Pt8PI\nOq6jJwEuLi5wcnJq1X6dnJw4SdWBtDrYPnbsGLZt24Zx48ZZXbrYuHEjNm3ahNdffx3Hjx+Hr68v\nYmJicPXqVbsMmMgeWhNki5XUgPrsdlJSEn766Se8/vrrAOqDb9uyEbb7IyKqZzKZcPDgQRw5cqTd\n+zh16hTee++9LqlVXrx4MTw8PFBQUIDZs2fDy8sLvr6++Mtf/iJ//vfffy+z/M8++yyUSiWUSiXu\nu+8+uY/g4OAG+163bh2USusQbOjQobJV7OHDh3HTTTcBqO8AI/b73HPPNfl4ANi6dSuuu+46eHh4\nQKPRYNmyZXLSqfCb3/wGYWFhOH36NKZNmwZPT08EBgbixRdf7Mivi5rRqmC7tLQUCxcuRHJyMtRq\ntbzdbDbj1VdfxapVqzBnzhyMGTMGO3fuRFlZWZOBC1FP1lhwrNPpkJiYCL1e32QNtljAhoE1EVHj\n8vPzsXbt2g7v529/+xvOnDnTCSNqmclkQmxsLAYNGoSXX34ZUVFRePnll/Hmm28CqC+n2bJlCwDg\nzjvvxO7du7F79248+OCDch9N1Vbb3m5Zhz169GgZWD/44INyv3feeWeTj1+/fj0efvhhaDQavPTS\nS5g/fz7efvttTJs2DdXV1VaPKy0txYwZM3D99ddj06ZNGDVqFFauXInU1NT2/qqoGa3qRrJ06VLc\nddddiIqKsrqckZ+fj6KiIkyfPl3e5u7ujqlTpyIzMxNLly7t/BETdTGxkIOoyU5ISLBaAZIBNhFR\ny06fPo0LFy50eD9XrlzBqVOnMHr06E4YVfNqampw9913Y82aNQDq46GIiAhs374dy5Ytg0qlwty5\nc/HQQw9h3LhxjX4eNFVe0lzZia+vL2JjY/F///d/mDRpUov7LS4uxvPPP49bb70VH3/8sQzEr7/+\nesTHx2Pbtm1ydWOz2YwLFy7gnXfewcKFCwEA9913H4KCgrB9+3bExsa28rdDrdVisL1t2zbk5eXJ\nTLXlmZR40fj5+Vk9xtfXFwUFBU3u88SJE+0aLPUsvek45uXlAajPDABAbGysPMOPjY3Fhg0bAACp\nqam4ePEi8vLyetXz703Ppa/jsew9evKxDAoKgru7e6u3N5vN7arTbsqXX36JuXPnNlpK0dkeeOAB\nq+9vueUW7N692+4/ty0++eQT1NTUYPny5VZx2r333ounnnoKH330kQy2AUClUslAG6ivF7/pppvk\nZ2FfVFZWhpMnTzZ6X0hISIf23Wyw/e233+Lpp5/GkSNHZNG+2Wxu1SQAtqQhRyLO5Ddu3Ci/P3Dg\nAC5dumR1f2xsLM/6iYjayGQytbmlXnM+//xz1NbW2n2SoKura4OEolqtblAH3d3OnTsHABg5cqTV\n7UqlEiNGjJD3C4MHD26wDx8fH3z99df2G2Qf1mywffToUVy6dAljxoyRt9XV1eHzzz/HP/7xD3kG\nUFRUhMDAQLlNUVER/P39m9zv+PHjOzpu6kYi29Ibj+OcOXNkZ5EJEyZAr9ejsLAQer2+V5aL9OZj\n2dfwWPYejnAsq6qq2rS92Wzu1FZ11dXVqKur67T9NaWjicOmHt8VY29OY11PgI53VHFkXl5eTb7m\nSktLO7TvZq+/zJkzBydPnkR2djays7Px1VdfYfz48ViwYAG++uorhISEwN/fH4cOHZKPqaqqwpEj\nRzB58uQODYzInprqPiICap1Oh8jISMTHx3NhGiKiDlIqlbj22ms7bX9DhgyBi4tLp+2vI5oLyNVq\nNQwGQ4PbbTPNbd2vraCgIABoMHHUZDIhNzcXQ4cObfW+qPM1m9n29vaGt7e31W0qlQpqtVpOTHj8\n8ceRmJiIUaNGISQkBOvXr4eXl1evzAJS7yLa+AlxcXEyALcMsPm3TETUMQqFolMz9dHR0U1mZztT\nawJesd7Czz//3OC+ESNGoLS0FN988w3Gjh0LoH7l4Q8++KDFfXt6eja5X1vTp0+Hq6srXnvtNdx+\n++1y33v27MHFixdxxx13tLgPgCXA9tKqbiSWbJcIXbFiBSorK5GQkICSkhJMnDgRhw4dkn8kRD2R\nZQYbsA6uLTuNEBFRxykUCoSEhEChUHRKqcLo0aO7JDBszVg9PDwwZswYvPvuuwgNDcU111yDYcOG\n4aabbsL8+fOxcuVKzJkzB4899hjKy8uxdetWjBw5Ev/73/+a/VnDhw+HWq3Gli1b4OnpCS8vL4wd\nO9aqtFcYMGAAnnnmGTzzzDOYPn06Zs2ahby8PCQlJeH666/HkiVLWvW8+nIZiT21OdhOT09vcNva\ntWs7pXcmUVexzWCzjR8RkX2NGTMGM2bMwEcffdSh/YwfPx7XXXddJ43KmmUAb5tcbO727du347HH\nHsOTTz4Jo9GIxYsX46abbsI111yDDz74AE888QRWrFiBYcOG4W9/+xv0ej2+/PLLJn82UN8hZNeu\nXVi1ahUeeeQR1NbWYu3atTLYtt3+6aefxsCBA7F582b8+c9/hlqtRnx8PDZs2GBVctOW50WdQ2Hu\notMYy+Jy29IUciyOMHmnJVqtFsnJySgsLIRGo0F8fHyfDLR7w7GkejyWvYcjHMuqqqo2tf4D6rOm\nmZmZiI6ORk1NTbt+rlKpRFpaGqKjoxkYUqdq7m+6ozFsmzPbRI7IdjJkXFwckpOTYTAYoNFoumlU\nRER9h0KhwM0334y///3vePjhh9u1j+effx6TJk1ioE0OhcE29XparRaJiYkygy3qtMXXrNEmIuoa\nzs7OmD9/PioqKvCXv/ylTTXCzzzzDJYsWQIPDw87jpCo89l/6SWibqTVaqHT6aDRaBAaGoq4uDhE\nRkY2OjGSiIjsT61WY9myZUhNTUVwcHCL22s0Ghw4cABPPvkkfH19u2CERJ2LmW1yeKJEpKnsdGRk\nJJKSkuT3ltsxo01E1PU8PT0RExODTz/9FNnZ2di/fz8+//xz2X968ODBuOWWW3DXXXchPDwcw4cP\nZ+kIOSwG29QniZ7aWq2WATcRUTdQKBQYOnQogoKCcMcdd6CoqAhXrlwBAPTr1w/+/v5wcnJikE0O\nj8E2OTzbYLmxTHdL2W8iIuoeCoUCTk5OCAgIQEBAQHcPh6jTsWabHJ7l0uuiRlt8DTQdYMfFxTH4\nJiIiIrtisE29huidLSQnJ8vAm4E1EVHn4mqD1FvY+2+ZwTY5NNvsdWhoqOwwYvk1ERF1HldXV1RV\nVTHgJodnNptRVVUFV1dXu/0M1mxTryCCbtF1RKvVsn82EZGdKJVKuLm5wWg0dvdQrOTm5gIAQkJC\nunkk9crKygAAXl5e3TwSao6bmxuUSvvlnxlsk0Njv2wiou6hVCrbvGS7PYk5O5GRkRg7dmx3DwcA\ncPLkSQDA+PHju3kk1J0YbJNDamz59ea+JyKi3qOpCfDNXdFkVyrqLgy2yWFYvlGKjLblYjVNbUtE\nRL2feL/n+z/1NAy2yeFotVro9XqEhoZ291CIiKgbNJe9FqUkrX0Mkb0x2CaHYJmp0Gq1LXYa4Zsq\nEVHfYVlayMnx1NMw2KYeqbnLgHwTJSKixrRmRWGirtZin5OkpCSEh4fD29sb3t7emDx5Mg4ePCjv\nX7x4MZRKpdW/yZMn23XQREREREJTC5fpdDo5x4eou7SY2R4yZAheeOEFhISEwGQyYceOHZg9ezaO\nHz+O8PBwKBQKxMTEYNeuXfIx9mwMTn1DS1kIZimIiKglbAtLPUGLwfbMmTOtvl+/fj22bNmCL774\nAuHh4TCbzXB1dYWvr6/dBkkkarW1Wi0DbSIiahPbeT+W+JlC9tam5XLq6urw7rvvoqqqClOnTgUA\nKBQKHDlyBH5+fhg5ciSWLl2K4uJiuwyW+h4RYBMREXWU6FZC1JVaNUHym2++waRJk2A0GuHh4YF9\n+/Zh5MiRAIDY2FjMnTsXwcHByM/Px5o1azBt2jRkZWWxnIQ6LDk5GUDT9XhERERNaWzCJLuVUFdT\nmM1mc0sb1dTU4Mcff0RpaSnef/99bN68Genp6Y0uP1pYWIigoCC89957mDNnjry9tLRUfp2bm9tJ\nw6febuPGjQCAlStXdvNIiIiIqC8KCQmRX3t7e7f58a3KbLu4uGDYsGEAgBtuuAHHjx9HUlKSzDpa\n0mg0CAwMxNmzZ9s8GOrbUlNTAdRfLREYZBMRkT009plDZA/t6rNdV1cHk8nU6H3FxcU4f/48NBpN\nk49vLCNOjuPEiRMAOv846vV6q/2yP6r92etYUtfjsew9eCy7hu1njj3wWPYOltUZ7dFisP3UU0/h\njjvuQGBgIMrKyqDVapGRkYHU1FSUl5dj7dq1mDdvHvz9/fH9999j1apV8PPzsyohIWoNBtVERNRV\nmlvyvbn7idqqxWC7qKgICxcuxIULF+Dt7Y3w8HCkpqYiJiYGVVVVOHnyJHbt2gWDwQCNRoNp06Zh\n//798PT07Irxk4NqzZsZ3+iIiIjI0bUYbDdWly24u7vLmiciIiKi3oRZbuoM7arZJmqrxt6wbHud\n8s2MiIh6Ctue3PyMovZisE09BjMIRETU3cRnkOjJTdRRDLapS9gG0AyoiYioJ7FN+PBzijoLg23q\ncuINTVyeE5kDvrEREVFPwCut1JkYbFO3YZBNREQ9RWs/ixiIU1spu3sA1PeINyjbCZJERERdTavV\nygBaiIuLYzBNnYaZbSIiIqJWYhBObcVgm7oFswZERNQTtPRZxLIR6iiWkVCXaOwyHRERkaPg5xi1\nF4NtsgvbNyWdTscabSIicjiNXYll4E1twTIS6hJcGICIiBwZy0iovZjZJruwzQTYfs+sABEROSrL\nVSaJWsJgm4iIiIjITlhGQt2Cl+OIiMhRsUMJtQUz22Q3CQkJSEhI6O5hEBER2RVLI6k5zGyT3TED\nQOElxgAAABSZSURBVEREvQk/z6gtGGxTp7IMrJOSkqxuIyIiIuprGGyT3TEDQERERH1VizXbSUlJ\nCA8Ph7e3N7y9vTF58mQcPHjQapt169Zh8ODBUKlUiI6OxunTp+02YOrZuAw7ERH1Nfzso+a0GGwP\nGTIEL7zwAr788ktkZWVh2rRpmD17NrKzswEAGzduxKZNm/D666/j+PHj8PX1RUxMDK5evWr3wRMR\nERER9WQtBtszZ87EbbfdhmHDhmHEiBFYv349vLy88MUXX8BsNuPVV1/FqlWrMGfOHIwZMwY7d+5E\nWVkZ63T7EM7CJiIiImpcm1r/1dXV4d1330VVVRWmTp2K/Px8FBUVYfr06XIbd3d3TJ06FZmZmZ0+\nWOoZUlNTkZqa2t3DICIiIurxWjVB8ptvvsGkSZNgNBrh4eGBffv2YeTIkTKg9vPzs9re19cXBQUF\nnT9a6nHY1o+IiIioaa0KtkeNGoWvv/4apaWleP/99zF//nykp6c3+xiFQtHkfSdOnGjbKKlHiY2N\nBVB/HPPy8uTX5Lh4/HoPHsveg8ey9+CxdGwhISEdenyrgm0XFxcMGzYMAHDDDTfg+PHjSEpKwv/9\n3/8BAIqKihAYGCi3Lyoqgr+/f4cGRo5BBN5ERERE1FC7+mzX1dXBZDIhODgY/v7+OHToECIiIgAA\nVVVVOHLkCF566aUmHz9+/Pj2jZa6XGNlIuIMncfR8fFY9h48lr0Hj2XvwWPZO5SWlnbo8S0G2089\n9RTuuOMOBAYGyi4jGRkZcoLc448/jsTERIwaNQohISGyWwlreHsfEXiHhoZ280iIiIiIHEOLwXZR\nUREWLlyICxcuwNvbG+Hh4UhNTUVMTAwAYMWKFaisrERCQgJKSkowceJEHDp0CJ6ennYfPNlfXFwc\nW/sRERERtVOLwXZycnKLO1m7di3Wrl3bKQOinktcreBEDyIiIqLWaVfNNvUtLAkiIiIiap82LWpD\nRERERB3HEs2+g8E2tYhvCERERJ1Hq9VCp9N19zCoizDYpkYxwCYiIrKfyMhIlmn2EazZphbxzYCI\niKjz8HO1b2GwTY0uXMM3AiIiIqKOYxkJAQB0Oh3LRoiIiIg6GTPb1GQWu7GMNxERERG1HoNtAsCA\nmoiIiMgeGGxTkxiAExERdS1eVe59WLNNRERERGQnzGwTERER9RDMaPc+zGwTEREREdkJg20iIiIi\nIjthsE1EREREZCcMtomIiIh6IK1WywXnegEG20RERERdgMFz38RuJEREREQ9EDuT9A4tZrY3bNiA\nCRMmwNvbG76+vpg5cyZOnTpltc3ixYuhVCqt/k2ePNlugyYiIiJyNHFxcQyg+6AWg+2MjAw88sgj\nOHr0KD799FM4Ozvj1ltvRUlJidxGoVAgJiYGFy5ckP8OHjxo14ETEREREfV0LZaRpKamWn2/a9cu\neHt7IzMzE7fffjsAwGw2w9XVFb6+vvYZJRERERGRA2rzBMkrV67AZDJBrVbL2xQKBY4cOQI/Pz+M\nHDkSS5cuRXFxcacOlIiIiIjI0SjMZrO5LQ+4++678d133+HEiRNQKBQAgPfeew+enp4IDg5Gfn4+\n1qxZg7q6OmRlZcHV1RUAUFpaKveRm5vbiU+BiIiIqPcT1QaxsbHdPJK+JSQkRH7t7e3d5se3qRvJ\nE088gczMTBw5ckQG2gBwzz33yK/HjBmDiIgIBAUF4aOPPsKcOXPaPCgiIiIiot6g1cH2n/70J+zb\ntw/p6ekYOnRos9tqNBoEBgbi7Nmzjd4/fvz4Ng2SOk709eyMWdAnTpwAwOPYG/BY9h48lr0Hj2Xv\n0dnHkn8T3cOyOqM9WhVsL1++HO+//z7S09MRGhra4vbFxcU4f/48NBpNhwZHREREROTIWgy2ExIS\nsHv3bnz44Yfw9vbGhQsXAABeXl7w9PREeXk51q5di3nz5sHf3x/ff/89Vq1aBT8/P5aQ9CDs60lE\nRETU9VrsRrJlyxZcvXoVv/3tbxEQECD/vfzyywAAJycnnDx5ErNmzcLIkSOxePFihIWF4ejRo/D0\n9LT7EyAiIiIi6qlazGybTKZm73d3d2/Qi5uIiIiIiNrRZ5uIiIiIiFqHwXYvo9VqZecRIiIiIupe\nDLaJiIiIiOykTYvaUM/HriNEREREPQcz20REREREdsJgm4iIiIjIThhsExERERHZCYNtIiIiIiI7\nYbBNRERERGQnDLaJiIiIiOyEwTYRERERkZ0w2CYiIiIishMG20REREREdsJgm4iIiMjBabVaaLXa\n7h4GNYLBNhERERGRnTh39wCIiIiIqGPi4uK6ewjUBGa2iYiIiIjspMVge8OGDZgwYQK8vb3h6+uL\nmTNn4tSpUw22W7duHQYPHgyVSoXo6GicPn3aLgM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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"from numpy.random import randn\n",
"import matplotlib.pyplot as plt\n",
"\n",
"N = 3000\n",
"a = np.pi/2. + (randn(N) * 0.35)\n",
"r = 50.0 + (randn(N) * 0.4)\n",
"xs = r * np.cos(a)\n",
"ys = r * np.sin(a)\n",
"\n",
"plt.scatter(xs, ys, label='Measurements', color='k', marker='.', s=2)\n",
"plt.scatter(sum(xs)/N, sum(ys)/N, c='r', marker='*', s=200, label='Mean')\n",
"plt.scatter(0, 50, c='k', marker='o', s=400, label='Intuition')\n",
"plt.axis('equal')\n",
"plt.legend(scatterpoints=1);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see that out intuition failed us because the nonlinearity of the problem forced all of the errors to be biased in one direction. This bias, over many iterations, can cause the Kalman filter to diverge. But this chart should now inform your intuition for the rest of the book - linear approximations applied to nonlinear problems yields inaccurate results."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The Effect of Nonlinear Functions on Gaussians"
]
},
{
"cell_type": "markdown",
"metadata": {},
"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."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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qublZhw8f1h133CFJOnXqlD7++ONy+8f169draGhIfX195XXsfX19SiaTY1pEjrRu3bpq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\nSUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"from numpy.random import normal\n",
"gaussian = (0., 1.)\n",
"data = normal(loc=gaussian[0], scale=gaussian[1], size=500000)\n",
"plt.hist(2*data + 1, 1000);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is an unsurprising result. The result of passing the Gaussian through $f(x)=2x+1$ is another Gaussian centered around 1. Let's look at the input, nonlinear function, and output at once."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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RLKrpplzIP4udxzZYxO4ccB/USk0DW9CN0rt2wISh063iZVUG5BRkOj4hIiIiumEsqumG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\nwevzQIIEKBH8ahhVoQGQYcpo8zmNuiQ8OiAbSYbWza1IxH/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SKZWXl+MPf/gD9u3bB7vd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\npI0oqSaEEEIIIaSN/h81dAAajRGj4QAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from nonlinear_plots import plot_nonlinear_func\n",
"\n",
"def g1(x):\n",
" return 2*x+1\n",
"\n",
"with book_format.figsize(y=5):\n",
" plot_nonlinear_func(data, g1, gaussian)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> I explain how to plot Gaussians, and much more, in the Notebook *Computing_and_Plotting_PDFs* in the \n",
"Supporting_Notebooks folder. You can also read it online [here](https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb)[1]\n",
"\n",
"The plot labeled '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 on the bottom lweft\n",
". 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. For the output I computed the mean by taking the average of all the points, and drew the results with the dotted blue line. A solid blue line shows the actual mean for the point $x=0$. 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 The computed mean, represented by the dotted blue line, is nearly equal to the actual mean. If we used more points in our computation we could get arbitrarily close to the actual value.\n",
"\n",
"Now let's look at a nonlinear function and see how it affects the probability distribution."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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k6r30nDTsOLYWO46tRQNNIzipXRAa0BaDu74IBX/DQ0RENk7WZcoTExOteWiL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\nT5/Ou+++y3PPPce0adNo1KiRhyMWQgghKsbhdGBxzVt9fe7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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def g2(x):\n",
" return (np.cos(3*(x/2 + 0.7))) * np.sin(0.3*x) - 1.6*x\n",
"\n",
"with book_format.figsize(y=5):\n",
" plot_nonlinear_func(data, g2, gaussian)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This result may be somewhat surprising to you. The 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",
"\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",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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21s3a462nluKvmD9w5NwupGvTTNoUFRfgbOIxnE08BgBQqxzR1r8jQlt0R2iL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\npI7+H3gQzE18m0bYAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y = g2(data)\n",
"gaussian2 = (np.mean(y), np.var(y))\n",
"with book_format.figsize(y=5):\n",
" plot_nonlinear_func(y, g2, gaussian2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As you can see the probability function is further distorted from the original Gaussian. However, the graph is still somewhat symmetric around x=0, let's see what the mean is."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"input mean, variance: -0.0017, 0.9984\n",
"output mean, variance: -0.1220, 2.4009\n"
]
}
],
"source": [
"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))"
]
},
{
"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",
"\n",
"$$m=\\frac{-3-3}{2-(-2)}=-1.5$$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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l2dh5fC12Hl8LV4UHrOW2CPIJxZiBU2HGT3iIiMjISbpMeWpqqiF3rReiKOLk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\nymsIv5+LNbTjAeCi5LIhqEr/Xl9muqyP7z+JuebVt3ihfsb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H89NPP2E2m1myZAlZWVll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\nRDVJUi2EEEIIIUQ1SVIthBBCCCFENf0/TTCoMuW7JmcAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"output mean, variance: 0.0025, 2.2465\n"
]
}
],
"source": [
"def g3(x): \n",
" return -1.5 * x\n",
"\n",
"with book_format.figsize(y=5):\n",
" plot_nonlinear_func(data, g3, gaussian)\n",
"out = g3(data)\n",
"print ('output mean, variance: %.4f, %.4f'% (np.average(out), np.std(out)**2))"
]
},
{
"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",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"linear output mean, variance: 0.1453, 7470.2044\n",
"nonlinear output mean, variance: -8.8565, 30504.9480\n"
]
}
],
"source": [
"out = g3(data)\n",
"out2 = g2(data)\n",
"\n",
"for i in range(10):\n",
" out = g3(out)\n",
" out2 = g2(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))"
]
},
{
"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.\n",
"\n",
"And, of course, we have minimized the issue by using a function that is quite close to a straight line. What happens if our function is $y(x)=x^2$?"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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7ekxvnWMX9+PS9TMYHDkKCb1Gw0puLUFSIiKi+yfpbcrT09NNuWuDUGvU2HLq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\nSUVORK5CYII=\n",
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""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def g3(x): \n",
" return -x*x\n",
"x0 = (1, 1)\n",
"data = normal(loc=x0[0], scale=x0[1], size=500000)\n",
"\n",
"with book_format.figsize(y=5):\n",
" plot_nonlinear_func(data, g3, gaussian=x0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Despite the curve being quite smooth and reasonably straight at $x=1$ the probability distribution of the output doesn't look anything like a Gaussian and the computed mean of the output is quite different than the value computed directly. This is not an unusual function - a ballistic object moves in a parabola, and this is the sort of nonlinearity your filter will need to handle. If you recall we tried to track a ball in the previous chapter and failed miserably. This graph should give you insight into why the filter performed so poorly."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## A 2D Example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is somewhat hard for me to look at probability distributions like this and really reason about what will happen in a real world filter. So let's think about tracking an aircraft with radar. The aircraft may have a covariance that looks like this:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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z6LYxfBMdUX95GmXX62KQbMM0SnjXMWPS5RERTczuanddXsBStYn1RROmxrML\naH8wfBMdMVmWY3voYqvvoOsO0PN7KJVTNBsifIcbh4jo6ArCFO3euLd73TyJpZqB4y0eD0/7i+Gb\n6Ii4PHT3vCG6XhflSoqlpgRV5hHIRHR0ZVmOziDAyEnRVFpo6XUcb3G1mw4GwzfRnMuyHJ2hO57T\nzdBNRHQFx4vR7vlQygZO1VpYaZhYaXCTOR0chm+iOZXnOTpDbxy63SG6fhfFUsLQTUQEIEkzbPUC\nBEEBy9oxNHUT64tVKBJPqaSDxfBNNGfyfDy9ZLPnoO8N0fHGobvVEKEp/BUqEdHQjtAZBKiKdazV\nm1hrmmjV1EmXRUcEwzfRnNgN3e2+g75roeN1gGKMVkNi6CYiAhBGKTa7PpAKOG6cRMscb6gUKqVJ\nl0ZHCMM30YzL8xy9kT8+Bt4boet2kBcjLNQl6CpDNxFRnufoDkMMRjGaShOt2gKOtQzUdLbg0eFj\n+CaaYT3Lw2bfQc+10HW7yAohmnUJuqpPujQioqngBQk2uz7EoopT1WNYrptYXdA5PpAmhuGbaAb1\nd1a6e+4IHbeDtBCiWRdhMHQTEQEA0izHds+H6+VY0lbQ1Gs4vmhCk3meAU0WwzfRDBnYPja6Nvqe\njY7bQYIACzURpsbQTUS0y3IibPcDGEIN72o0sdIwsFTXUChwfCBNHsM30QwY2P54eok7QsftIso9\nLFQlVHWGbiKiXVE8PqEyjcs4pq+jaZhYXzQhCow7ND34r5Foig2dABtdGwPXRsfrIMzGodvUdK7g\nEBHtyPMcPStE34rRkBew2FjAWtNAw1QmXRrR2zB8E00h2wtxsTNC39kJ3amHhZqENYZuIqIrOF6M\nrX4AsaDgZHUNSzUTa00DZW6opCnF8E00RfwwxsXOCN2Rg21vG0HioFEVsaYzdBMRXS6KU2z1AkRR\nEYvaKhZUE8cXTeiKOOnSiK6J4ZtoCkRxikvdEbYtB123g1FkoVEVsWowdBMRXS7LcnQGASwnwYLS\nxHqjjtUFA82qwudLmgkM30QTlKYZNvsOtgY2um4Pg6APQyvjjkUdpSJfRIiILje0I3SHIdSyjjtq\nLSxWdayyxYRmDMM30QRkWY7toTue1e310fN7UOQCTqyqqJT5IkJEdDk/SLDVD4BUwJp2HAu6gWMt\nE4pUmXRpRDeN4ZvokPUsDxs9G11niI7XQUVIsbYkQxJKky6NiGiqJEmG7UEAz8vRVBfR1GpYXTBQ\nN3gsPM1UzvsKAAAgAElEQVQuhm+iQ2I5AS51x6dSbjtbyIsRlpoyVJkvIkREl8vzHP1RhP4wginW\n8K7GApbr44NyimzJoxnH8E10wFw/wqWuja5tY9vdQph5aNUlHgVPRHQVl48OXDdX0TJ1rDUNHpRD\nc+OazaXf+MY3cN9998E0TbRaLTzwwAN45ZVXrnunL7/8Mj70oQ9BURSsra3ha1/72r4VTDQrwijB\nGxsD/N9zbfzv9jm8OfozVC3FHWs6DFWYdHlERFMljFJcaLvY6iZYVFZxunkSd59Ywh2rdQZvmivX\n/Nf84osv4tFHH8V9992HLMvw1FNP4aMf/SheffVV1Gq1q95mNBrhYx/7GD784Q/j7NmzeO211/D5\nz38eqqriiSeeOJC/BNE0SdIMG117PMHE62EYDlAzKjjFCSZERG+TZjm6HB1IR8g1w/dzzz13xcc/\n+tGPYJomfvOb3+CTn/zkVW/z4x//GEEQ4MyZMxBFEXfddRf+8Ic/4Dvf+Q7DN821LMuxNXDQ7jvo\nen30vC40pYhTqyrKnGBCRPQ2fzk6cKlmYGVB5+hAmms39a97NBohy7J3XPUGgN/+9rf44Ac/CFF8\n64Spj3/849jY2MD58+dvvVKiKZXnOTpDF//3z9t47dIl/G/vj/CyHo4vy1huKgzeNDGvvaZMugSi\nq/KDBOc2HAyHwJp2HKeb67j7xBKOL5oM3jT3bqqJ6rHHHsP73/9+3H///e94nXa7jePHj19x2eLi\n4t7n1tfX33abs2fP3kwZNKWO4uNo+zG2rQDDwMEwHKBQiVHTy4iEIkb9SVd361597dVJl0D74LXX\nVD6Wc2QeHsskzTEYJQjCImpCDTVJRxC2Yft9vLo96eoOz1F8vZwnp0+fvq3b33D4fuKJJ/Cb3/wG\nv/71r6/Zg8X+LDoKgjjF1tDH0PPQjwZICgFq1RJUiRspafJee03Ba6+p+M//rwkAuPNOF3fe6U24\nKjrK0iyH5aSw3QxmpYqmamBBl9HQRY4OpCPnhsL3448/jmeeeQYvvPACTpw4cc3rLi0tod1uX3HZ\n1tbW3ueu5t57772RMmhK7f4P/ig8jrubKduDESShAzXOsV6toqoLc/Efz92VtbvuvGvCldDtuOvO\ntx7LL/2/TQDNyRZEt2WWfy6zLEd/FKJvRXhXvYqmuoBWVcdKQ4dQOXoHix2l18t5ZlnWbd3+uuH7\nsccew7PPPosXXngB7373u697h/fffz+++MUvIgzDvb7v559/Hqurq1dtOSGaFZ2hi42ujW23h57X\nhaGVOMGEptqdd7pg8KZJGdoRuoMAUlnDCXMNTVPH6oIOWeSR8HS0XXNXwyOPPIIf/vCH+PGPfwzT\nNNFut9Fut+G67t51nnzySXz0ox/d+/jBBx+Eoih4+OGH8corr+A//uM/8M1vfpOTTmhm2V6IV891\n8NrFTfxv709w0i6OL8tYbMgM3jTV2GpCk+B4Md64aMOyCljT1/Hu5gncfWIJ71qtM3gT4Tor39//\n/vdRKBTwkY985IrLv/rVr+Kpp54CMN5E+cYbb+x9zjAMPP/883jkkUdw7733ol6v4wtf+AIef/zx\nAyif6OBEcYqLnRG2rRG2nC0EmYulhgxNkSZdGhHR1PGDBNuDAGlSRktZQUOrYnVBR02XJ10a0VS5\nZvjOsuy6d/D000+/7bK7774bL7744q1XRTRBWZaj3Xew2R+h43YxCAaomxWsmvpc9HUTEe2nKE6x\nPQgQ+MCC0sKCWcNyQ+chOUTvgOe1El1mYPu42Bmh6wyx7W5BUQo8JIeI6CqSNEN3EMJ2U9TlBo41\n6liq61isqShxVjfRO2L4JgLghzEubI/QGY3QdtrIiyFWF2XIEn9EiIgul2U5+laIwSiGIVZxR20B\nraqGlQUdlfLRm2BCdLOYLOhIu3x0YMftwElGWKiKqOr6pEsjIpoqeZ5jaEfoWRGUso4T1TU0DR2r\nTQOSwDhBdKP400JH0vhIeA+bvStHB55c1DjBhIjoL4zcCJ1BCKEgY01bx4I+Dt2azIPFiG4Wwzcd\nObYX4sL2CF3Hwpa7hXIlwfFlGaLAX5cSEV3OCxJs9wPkaRlL6hoaqoHVpoGqxqlPRLeK4ZuODI4O\nJCK6MWGUYrsfIIwKaClLaKhVrCzoWDCVSZdGNPMYvmnu5XmOzd5bowOHwQA1jg4kInqbMErRHYbw\ngxx1uYH1Rh1LdQ2LNQ1FtuQR7QuGb5prthfizS0LHWeALWc8OvAkRwcSEV0hiseh2/Uy1OUGVus1\ntKoalhs6yhwbSLSvGL5pLqVphoudEdqD8ejAKPewsihD4ehAIqI9cZKhOwjgeBlqch3L9fpO6NY4\nNpDogDCJ0Nzpj3xc2LZ2pph0dlpMNLaYEBHt2A3dtpeiLtXxrnoDraqGpboGocLQTXSQGL5pboRR\ngje3LWxbI7SdTRTKMdZXFL6QEBHtSJIM3WGIkZugJtXwrlpjr72Ez5VEh4Phm2ZenufYGri41LWw\n5XQwigZo1iRUdW3SpRERTYUkydC1QthOAlMch+5mVcNyXYPIA3KIDhV/4mimuX6E81sWOs4QbacN\nRQZOrmrcIEREhPEpvr1hCMtJUBVrOFWro2mOV7p5KiXRZPAnj2ZSluW41B1hsz9C29mCn9pYbipQ\nZf6TJiK6PHSbYhWnag00DQ0rCwzdRJPGn0CaOUMnGG+otHvoeB2YegnLVZ0zaInoyEvSDH0rxNBO\nYAgmTlUbaJo6lhsaZLEy6fKICAzfNEPiJMWbWxa2rBE27U2gFOHYsgyJx8IT0RGXZvk4dI9i6IKB\nU9UFLBjj9hJFYugmmiYM3zQTtgcuLnVH2HY6GAR9LNQE1AxuqCSioy3NcgysEP1RBEMwcaK6hgVD\nxwpDN9HUYvimqeaH8XhDpW1h096EKGU8oZKIjrwsy9EfhRiMImgVAyera2joGlYaOlRZmHR5RHQN\nDN80lbIsx2bPxkZvhC13C248wtKCDE3hSg4RHV27K90DO4JaMbBu7ITuBR0aQzfRTGD4pqnj+BHO\ntYfYtvvoeNvQ1CJOLXJDJREdXUmaY7vvY2iPe7p3Q/dyQ4OuiJMuj4huAsM3TY08z3Gpa2OjN24x\niXIPa4sKJJEbKonoaIqTDF0rgefnaDVUnKo2UNdVLDe40k00qxi+aSp4QYw/bw6wbQ+w5bZRNcpY\nrWooFLjaTURHTxil6FkhHC9FKdaxKuu4c/kEluoaN1ISzTiGb5qoPM+x2XNwqWth02kjyBwcW+Jq\nNxEdTUGYojsM4Ac5anIdp+t1qHYRDV3EqZXapMsjon3A8E0T44fxuLd7NEDbbUNXizhV52o3ER09\nrp+gZwWIwgIaSgPHGjU0qyoWaxpgb066PCLaRwzfdOjyPEe77+Bix0Lb3YKXjLDSUqBI/OdIREeL\n48XoDkOkSQkNuYX6QhWtqorFuoZyiSNVieYR0w4dqjBK8Of2EFujAdr2JjSNk0yI6OixnAi9YYhC\nXsGCsoRa1cRiXUPTVFBi6CaaawzfdGg6Qxdvbg/RtrfgxBaWWwpUmf8EiehoyPMclhOjOwxQKcho\nKauoKwYWaxoWTIWLEERHBJMPHbgkzXCuPcTWcIhL9gZkOcfJRR0lvtAQ0RGQZTkGdoS+FUIqKVjV\njqOm6Fisa2gYMve5EB0xDN90oIZOgPPtIdr2NgZhD4sNCYbK2bRENP8uP41SKes4pi+jrmlYqmuo\n6fKkyyOiCWH4pgORZTkubFvY6FvYsDdQrEQ4uaKhXGYvIxHNtyTJ0LNCWM5bp1HWNRVLdQ2mJk26\nPCKaMIZv2neuH+1squyi422jURVQN7VJl0VEdKCiOEXfijByExiCiVPVBhrGeKWbp1ES0S6Gb9pX\nmz0bFzpDbNqbiOHh+LICUeCBOUQ0v1w/wWAUjg/GkWq4o1bDwk7o5mmURPSXGL5pXyRphv+90EPb\nGmDT2YChFbFa44E5RDSf8jzHyI3Rt0JkaRkNeQHHGlUsmCpaNRWSwJdXIro6PjvQbfPCBJd6Hizh\nEoZRH8tNmSMEiWgupVmOoR1hYIUQigqayipqsoFmVUGzqvJgHCK6LiYkui3tvoM/b4+wHXQgZiWc\nWFa5qZKI5k4Up+iPIoycGJpgYE1fRk3VsFhTUee4QCK6CQzfdEvSndndm8MBLnkb0NQc68vcVElE\n88UPEvRGIXw/R1Wq4lS1hrquYrGmwVDFSZdHRDOI4ZtumhfEeGNzgA1rG/2gg0a1AEXipkoimh8j\nN0LfipAmJdTlBlbrVTR3+rllkZsoiejWMXzTTekMXZxrD7BhbyAp+DixouF1l20mRDT7sp1+7r4V\nolKU0ZCXUKuaaFZVNKsKKmUuMhDR7WP4phuSZTnObw2x0R/gon0JmlrAal1lnyMRzbw4ydDfORRH\nrehY05dQVTQs1jXUdRnFIp/niGj/MHzTdflhjDc2BmjvHJqzuCDyiHgimnlBmKJnhXD9FKZYxclq\nDXVtvImSJ1ES0UFh+KZr6lkezm0NsGG3EaQ21lcUCBX+6pWIZpftxuhZIZKkiLpcx0p9Zz53VeWh\nOER04Bi+6aqyLMeFbQuX+kNcsi9BljKcWNT461cimklZlsNyIvSsEGVIqMtLqFdNLJgKWjWV/dxE\ndGgYvultwijBnzYGaI962HbbaNZFVHX+CpaIZk+cZBiMxv3cclnDqraIqqJjsaaiYShcUCCiQ8fw\nTVewnGA8RnC0CTcd4diyAkngihARzRbHizEYRQjCHKZYxQmzipqqYrGuocp+biKaIIZv2rM9cHFu\nq48L1gVUxBQn2WZCRDMkzXJYdoSBHaGYV1CTmjjeMFE3FPZzE9HUYPgmAMCFbQtvdvq4OLqAqlnC\nQlWZdElERDckiFIMrBC2l0ITdKyo49aSpqmgYSool3gWARFNj+s+I/3yl7/EAw88gLW1NRSLRZw5\nc+aa1z937hyKxeLb/vz85z/ft6Jp/2RZjj9e6uON7W1cGJ1Ha6GChSp/JUtE0y3Pxxsoz204uLgZ\noJJXcap2B/5q8QT+n/VV3H2yhcW6xuBNRFPnuivfruvive99Lz73uc/hoYceuuFDVX72s5/hnnvu\n2fu4VqvdepV0IOIkxR8v9bEx7KDjb2FtUYEs8ZchRDS9kiTDwI4wtCOIRQUNeQnVnaklTVOBKPA5\njIim23WfpT7xiU/gE5/4BADg4YcfvuE7rtfraLVat1wYHSw/jPHHS31cGG7CSYZYX1Y5v5uIppbr\nJxjYITw/gyGYOG6soqaqaFZVnkJJRDPlwJYIPvWpTyEIApw+fRqPP/44Pv3pTx/Ul6KbtDvR5E3r\nItKChxMrGkp84SKiKbM7m3swioC8grq0gNW6iYahoFlVock8aZeIZs++h29d1/Htb38bH/jAB1Au\nl/GTn/wEn/nMZ3DmzBl89rOf3e8vRzepM3Tx5/Z4ookgpVhdUG+4lYiI6DCEUYqBHWHkxFAqGpbU\nRVRlDQumggVT4YE4RDTTCnme5zd6ZV3X8d3vfhcPPfTQTX2RRx99FL/61a/w+9//fu8yy7L23n/9\n9ddv6v7o1mwNfbQtB1vBFnQ1R1VnbyQRTYc8z+GFGUZuijguwigb0CsadFlETRWhy2UuFBDRVDh9\n+vTe+6Zp3vTtDyV93XffffjBD35wGF+KriLLcmz0PWy7Nrb9LTSqJWgygzcRTV6a5bDdFLafoZyL\nMIQadE2DqQioaQJE7kUhojlzKAnsd7/7HVZWVt7x8/fee+9hlHEk7U400cUuAj/Dna27oezzRJNX\nX3sVAHDXnXft6/3S4eNjOT+m/bH0ggSDUQTXT9GsG6hKNVQVFa3aeANliSMC95w9exYAXyvnAR/L\n+XB598atuKFRg7ttIVmW4fz58/jd736HRqOBY8eO4cknn8RLL72EX/ziFwCAM2fOQBAEvO9970Ox\nWMRPf/pTfO9738O3vvWt2yqUbt7uRJOLwzbsZMCJJkQ0UbsnUA7tCHlWRk2uY7lmoq4raNVU6Io4\n6RKJiA7cdcP3Sy+9hP/zf/4PAKBQKOArX/kKvvKVr+Dhhx/GD37wA7Tbbbzxxht71y8UCvj617+O\n8+fPo1Qq4T3veQ+efvppPPjggwf3t6C38YIY/3uxizeHlxDDxfqyysMmiGgiLl/l1gQdS+oiTOmt\nDZRcFCCio+S64fvDH/4wsix7x88//fTTV3z80EMP3fSGTNpfu8H7/OACUA6w3uJEEyI6XEmawXJi\nDO0IhbyCqjRe5a7p48BtqiKfl4joSOKuuznzl8F7taXwBY6IDs3uYTiul8IQDayoizBlDQ2Dq9xE\nRADD91xh8CaiSUiSDENn3MtdgoCq1MBKfdzLvWAqMDVp0iUSEU0Nhu85weBNRIfN8cZtJV6QwRAM\nrGnLMGSVh+EQEV0Dw/ccYPAmosMSxSksJ4ZlRygXRFR3jnzfXeU2VE4sISK6FobvGcfgTUQHLc9z\njNzxKncYAqZk4pixClNW0DAVNAyZq9xERDeI4XuGMXgT0UEKwhRDJ8LIiSGXFdSlJRi6jvrO5klN\nFiZdIhHRzGH4nlEM3kR0ENIsx2hn82SallAVTZyqVmGq48Bd0ySePklEdBsYvmcQgzcR7TfXTzC0\nxwfhqBUNLbkFU9JQN2QsmApksTLpEomI5gLD94zxQwZvItofYZRi5I43T5YKAqpSHUs1EzVtHLir\nmsTnFyKifcbwPUPiJMUfL/VxfniRwZuIbkmSZhg5MUZujCQuwBBNHDNMGPJ442TD4EE4REQHieF7\nRuR5jj9tDHDJaiOBxyPjieiG5XkO24sxcmJ4QQZN0NGUWjCqGmqahAY3TxIRHRqG7xlxrj3E5rAH\nK+rjxIrG4E1E1xVEGdpdHyM3hlRSYEotrNZ1VDUZDUNmWwkR0QQwfM+Adt/BRn+ALXcTx5ZVlDlp\ngIjeQRSnGDkxLm5HKGRlrC9XcapqwFDGLSV1Q+ZzCBHRBDF8T7mhE+D8Vh8X7YtYbkqQBPZiEtGV\n0iyHvXMIThwDhmiiJSxDFST89epxNEwFksCneyKiacBn4ynmhzH+tNHDhdEF1MwyNIWjvohoLM9z\nOF4Cy43g+RnUioYFaQFmVUdVk6B6AVSpjNWmMelSiYjoMgzfUypJM/zxUh8XRxsQpQwNU5l0SUQ0\nBfwggeXGsN0YQlGGKTaxWjdgqhIaxng8YLFYQPcin96JiKYRn52n1J8u9d+abLKgTrocIpqgOMlg\nOREsJwayMkzJxAnT2BsPWDdkVMpsSSMimgUM31Noo2ujPRrAigY4scKRgkRH0W4ft+VECEPAEA2s\nqiZ0SUXdGE8r4amTRESzh+F7yjh+hIvdIdrOJlZanEpAdJRkWQ7Hf2set1JR0RAb0HUNNX08rcRQ\nxUmXSUREt4Hhe4qkaYY/bw6wYW/A1EtQJD48RPMuy3K4/njjpOunUMoKdKGJlZoOUx2vcNd0GcUi\nfwNGRDQPmO6myJvbFrbsLtJCgIUq+7yJ5tXupJKRG8P1E0glBYbYxHJNh6lIqOkyarrEPm4iojnE\n8D0lBraPzYGFrtfB+orCPm+iOZPn4xXukRvD8RKIJRmm2MBiTYepyKhp49AtVBi4iYjmGcP3FMiy\nHBc7I2zam2jWRb74Es2J3cBtuzEcP0GlIMGU6mjVDBiyhLouM3ATER0xDN9TYLNnY9vuA6UIVV2b\ndDlEdJt2A/fIjSEUJRhSHU1Thy6PxwLWNAkiT5wkIjqS+Ow/YUGUYKM3QtfvYG1RnnQ5RHSLvGDc\nUmK7MSoFEbpYxcmqDl3aCdy6zCPeiYiI4XvS3tyysO12oClFSCJ/9Uw0Sy4/bbJcEKCLJtYNA4Ys\n722a5CxuIiK6HMP3BFlOgM5oBDse4tSiPulyiOgGBGEKy4lguzGKBQGGYGDdMKHLb22aVCQGbiIi\nujqG7wlq9x30/B7qpogSZ/gSTS0/SGB74x7uQl6BIeo4ZpjQpfEKd52Bm4iIbhDD94TYXoi+48JL\nbCzrXPUmmia7U0ocbxy6SwUBuqBjTTOgSwpq+nhSiSoLky6ViIhmDMP3hLT7Drp+FzVD4Ml1RFMg\nzXI4Xgzbi+H5KYSiBF2sYt0YTympahKqmgSNgZuIiG4Dw/cEeEGMnu3AiUa4g73eRBOTJBnsncDt\nBxnUigpNqGKppsGQpb3AzU2TRES0Xxi+J6A38jAMhjC1Cnu9iQ5ZGKWwvfEpk1EEaKKGmtDAMVWD\nobwVuHnwDRERHQSG7wkY2AFGoY3VKn99TXQYdjdM2l6CPC1CE3Q0JR26qcJUx2HbVEWUSsVJl0pE\nRHOO4fuQ2V4IO3CBYgJJ5KE6RAfhnTZMrqo6NFHZW93WFZF7LoiI6FAxfB+y/siHFY5gqOwhJdpP\nf7lhUizJ0MUaTpgaNIkbJomIaDowfB8y24/gRA5W6wwARLeLGyaJiGjWMHwfojTN4EcxkjyGJLDl\nhOhm5XkOL0jh+uMNk0lS4IZJIiKaKQzfh8gLYwRJALHCTV1ENypOsp3+7RhekEIoitAEAyuqBlVU\nYCgiN0wSEdHMYPg+RF4wDt+SyBU5oneyu7rteDFcP0GaFqAKGvRKDUs1Fbo0DtqGKkKTBRQK3DBJ\nRESzg+H7EEVJiiiJIChcnSO6XJxke2HbDVKIReltq9uGKsJURVTK/M8rERHNLobvQ5RlOXJkHG1G\nR97uKEDXT+D4CbKd1W2j0sByTYUuizAUEaYmQZUqXN0mIqK5wfB9iPKdP8wRdBRFcbo3e9sLd1a3\nRROrqrq3ur3bTsLVbSIimlcM34coy3JkecbwTUfC3kE3OyvcWVqAJmgwKw2sKOPVbVOVYKgiV7eJ\niOjIYPg+ROVSEeVCGUkST7oUogMRhCncIIHnj1e3pZIMVTCxqmrQRBmGKu71b3N1m4iIjiKG70Mk\nCWWIZQFh7E66FKJ9EUTpOGgHCbwgRbkoQCkrqAoKVhQVhiLttZOoPFmSiIiI4fswyWIZYlmC5aeT\nLoXoloRRCi9Idla3U5QKFagVBUZFxVJNgSqK0GUB+s7qdplzt4mIiK5w3VfGX/7yl3jggQewtraG\nYrGIM2fOXPdOX375ZXzoQx+CoihYW1vD1772tX0pdtbJYgViSUQYZ8iyfNLlEF1XFKcY2hEubXt4\n/c0RLrRDBK4IvdjCqeoduLN1Gn+9egrvXT+G99+xgrtPtrC+VEXdkBm8iYiIruK6K9+u6+K9730v\nPve5z+Ghhx667qao0WiEj33s/2/vTmOjKt82gF9nX2amq3QDFMqfN4CoeQMhwS1IEMQA4gfBItGS\naGI0JggaIpYACaJ8waBAgiIFjNH4wUTUL62CGCIkrRaTWiX4Kv2DtWWRFtpZzpxznvfDtMPe2sU5\nXa5fMjmdZ5be5OmkF3ef55yHMWvWLNTW1uKXX37BihUrEAqFsGrVqgErfChSFRlZtgnrko32WBJZ\nIf4ZngYX1xOIJXw0nYsiGnMBocDWQwipOSjItmHrJiJ2qrMdsXQYOv94RkRE1Bs9/uacP38+5s+f\nDwAoLy/v8Q0/+ugjxONx7Nu3D4ZhYMqUKfj111+xdevWER++ASA/y0JOaw4uXmpm+KbAua6f3iDZ\nEXfRdM6DqZiwkY/bskKwdSMVtDsDt8mwTURE1C8D/pv06NGjeOCBB2AYRnps7ty5WLduHRobG3HH\nHXcM9LccUvKzbOTa2Th7/iw6Yi5CFsMMZY7r+anNkbHUWUk8V4Kt2whpWciLhOCGfYQMFXfdfgci\ntg7L0IIumYiIaFgZ8OTX3NyM22+//ZqxwsLC9GMjPXzLsoTC3BDaYoX46/yfKB0d4RUv6V8hhEDC\n8RFLpM5EEku48D0ZtmbB1rOQE7IRMiyELT29SVKPngcAFOSGAq6eiIhoeBrw8N2XC2XU1tYOdBmD\nmhACZ8+248/LLfjrz1PIzx4e3e+GXxqCLmFE83yBhCMQd3wkkj6cJKBChama0GUDhmzAVHV4egye\n4cA3LyOpKWhtldB63XuNtM/kcMa5HD44l8MH53JomzhxYr9eP+Cpr6ioCM3NzdeMtbS0pB+j1H9Q\ninNtRJ08/Bltgq55iNi84Aj1TiLpI+GI9NH3JRiyAUM2kaMaMDQdpqbB0hXYhgpTV2Bq/DkjIiIK\n0oCH75kzZ2LNmjVIJBLpdd/V1dUYPXr0LZecTJ8+faDLGBL+52IH/u+v8/jvpUaMytOQHR6aGzC7\nOt5TJk8JuJLhy/V8xBOpc2zHEh7ijo9sWYelmrBUG5ZmwdZN2IaGsKUjZGoIWXqvT/fX1Y0ZqZ/J\n4YRzOXxwLocPzuXw0NbW1q/X/6NTDZ48eRIA4Ps+Ghsbcfz4ceTn52Ps2LF47bXXUFNTg6+//hoA\nsGzZMmzcuBHl5eWoqKjAiRMnsGXLFmzYsKFfhQ5HBbkh+EJAQOD03/+FEEBOZGgGcBpYccdDrDNo\nR+MuPE+CpVqwtSzkGxaskIWQoSPUGbTDFjdHEhERDQU9hu+amhrMnj0bQGq5xPr167F+/XqUl5dj\nz549aG5uxu+//55+flZWFqqrq/Hiiy9i+vTpyMvLwyuvvIKXX3753/tXDGFFeWEIkbrgzpmLZxCL\nR1GYb3ET5gghRNeyEQ/xhIe44yHh+FA7u9q2aiE/YsPSDIRMvV9dbSIiIgpej+F71qxZ8H3/lo9X\nVlbeMDZ16lQcPny4f5WNIMX5EWiqAkPV8NflZvzRdBmjC2yYOtfnDiepDZGpkJ1wUkHbcQVUSYOp\nmjDVCPINA1bIgm1cCdrsahMREQ0fw+M0G8PAbdl2KmQ16Wi59DdO/9WMSEjBbTkGVJUdzqEm6frX\nhOy448HzJBiKAVM1YakmcmwDhmrC1jVYhgbb1GDpKrvaREREwxjD9yBi6iom33EbIud0hP8O4Xz0\nAn7/8yJyIiryc0woXIoy6HQtG7myZCTV2VYkFYZqwFTDyFJMFERMmFqqg20bGixD7TxqXGJEREQ0\ngsa4sJIAAArcSURBVDB8DzKSJGFsQTYKckJouhDC2dZcnI+ex2+n25Blq8iJ6LBMTlsQkm5qbbaT\n9FPd7IQHx/Whd3azDcVAxDBhhAxYmp4O2LaZCtm8NDsRERExDQxShq5ifHEuCnPDaLoQxoXLHWiL\nt6HpbCskOYaciI6skMYlKQPM8wUcx0Mi6afCdtKD4/hIegIKFOiKAV2xYKsm8kKp0G3qajpgd3W1\nNZXr9YmIiOhGDN+DnG1q+M/oPIx1snC+LRsXLo1CW7wdrdFWnL/YDk0TCFkqwrYGmx3xf8TzBZJJ\nH46b6mInk346bPu+BEPRoSk6DEVHlmJAD+nQVR2mpsLQ1CthW1e5bISIiIh6hWltiDB0FaNHZaHk\ntgjaOrJxoS0Xl2MO2hMdaHfa0XKuHUk/CttUUlcyNFK3kbhxz/cFXM+H64lUuHb9a45CSNAVDZqi\nQ5d1WIqGLDMVtnU1tTyk69YVtk1dZcgmIiKifmP4HmIkSUJO2ERO2IQQAu0xB20dCbS1x9GeSCCW\njCKWiOPvjjhibgcURaSCuKZAVWXomgxdlYfkcpWuQO26nUfvqqN75b4EGaqsQpXVzoCtIazo0C0N\nWliHoWowNAWGpqaOupq+r/Py60RERPQvYvgewiRJQsQ2ELENjBmVBSfpoT3mIJpIoqPzGE86iLkx\nOE4C0biLVtdB0o/Dhw9VkaCrMjRVhqbJ0FQJsiRBllNHSQJkufPYOd5XQgj4InUUItWdFgLwhbjh\n65sFbM8XUCQFiqxClVSoigZVUqErKmxZhWp1jatQZRmaqkBTZOiaku5gdwXtkfjXACIiIhocGL6H\nEV1TkKdZyIOVHoslkojGk0gkPSSSLpykl9pE6LpwvCSSfhJJz4ETSyLmJyGEn7rkvfDgQ8AX/lVj\nPtAVxCVAklPHpvMOJAmwm9qvhOvrQjYASJLcGeplyJIMGamvJUjpcUVKda0tWYWqqVCNVAdbkRVo\nipIK1eqVcH2z+1weQkRERIMVw/cwZ3WeS/p6vi/guB4SjtsZxj04SS/dfb718UoQ90XqfqvaAR8C\nBdaYdKC+MWRf6ahff7y6s67I8k3DtarIkCSGaiIiIhraGL5HKFmW0hsJe0PcLJS3tcD3gf8tHXvL\ncM3gTERERMTwTb0kSRIURcLV2xKtzgAftvRgiiIiIiIaIrjzjIiIiIgoQxi+iYiIiIgyhOGbiIiI\niChDGL6JiIiIiDKE4ZuIiIiIKEMYvomIiIiIMoThm4iIiIgoQxi+iYiIiIgyhOGbiIiIiChDGL6J\niIiIiDKE4ZuIiIiIKEMYvomIiIiIMoThm4iIiIgoQxi+iYiIiIgyhOGbiIiIiChDGL6JiIiIiDKE\n4ZuIiIiIKEMkIYQI4hu3tbUF8W2JiIiIiAZEdnZ2r1/DzjcRERERUYYwfBMRERERZUhgy06IiIiI\niEYadr6JiIiIiDKE4ZuIiIiIKEMyHr7HjRsHWZZvuC1YsCDTpVA/ua6LtWvXorS0FJZlobS0FOvW\nrYPneUGXRn1w+fJlrFy5EuPGjYNt27jvvvtQW1sbdFnUje+++w6LFi3CmDFjIMsy9u3bd8NzNmzY\ngNGjR8O2bTz00ENoaGgIoFLqSU9z+dlnn2HevHkoKCiALMs4fPhwQJVST7qbS9d1sWbNGtxzzz0I\nh8MoKSnBU089hdOnTwdYMd1KT5/LdevWYfLkyQiHw8jLy8OcOXNw9OjRHt834+H7hx9+QHNzc/r2\n448/QpIkLF26NNOlUD9t3rwZu3btwrvvvosTJ05g27Zt2LlzJ958882gS6M+ePbZZ1FdXY39+/ej\nvr4ec+fOxZw5c9DU1BR0aXQLHR0duPvuu7Ft2zZYlgVJkq55fMuWLdi6dSu2b9+OmpoaFBQU4OGH\nH0Z7e3tAFdOt9DSX0WgU999/P7Zu3QoANzxOg0d3c9nR0YG6ujpUVFSgrq4On3/+OU6fPo1HHnmE\njatBqKfP5aRJk7Bz507U19fjyJEjGD9+PObNm4eWlpbu31gEbNOmTSI3N1fE4/GgS6FeWrBggSgv\nL79m7OmnnxYLFy4MqCLqq2g0KlRVFQcOHLhmfNq0aaKioiKgqqg3wuGw2LdvX/q+7/uiqKhIbN68\nOT0Wi8VEJBIRu3btCqJE+oeun8urnTt3TkiSJA4fPpzhqqgvupvLLg0NDUKSJFFfX5+hqqgv/slc\ntrW1CUmSRFVVVbfPC3TNtxACH3zwAZYvXw7DMIIshfpg/vz5OHjwIE6cOAEAaGhowKFDh/Doo48G\nXBn1luu68Dzvhs+haZo4cuRIQFVRf/zxxx9oaWnB3Llz02OmaeLBBx/E999/H2BlRHS1rosO5ubm\nBlwJ9YfjOHjvvfeQn5+PadOmdftcNUM13VR1dTVOnTqF5557LsgyqI9eeOEFnDlzBpMnT4aqqnBd\nFxUVFXj++eeDLo16KRKJYObMmdi0aROmTp2KwsJCfPzxxzh27BgmTpwYdHnUB83NzQCAwsLCa8YL\nCgq4lIhokHAcB6tXr8aiRYtQUlISdDnUB19++SXKysoQjUYxatQofPXVV8jLy+v2NYF2vt9//33M\nmDEDd911V5BlUB+98847qKysxCeffIK6ujrs378fO3bswJ49e4Iujfrgww8/hCzLGDNmDEzTxPbt\n21FWVsa1pcMQ55QoeK7rYvny5bh06RIqKyuDLof6aPbs2fjpp59w9OhRLFiwAAsXLkRjY2O3rwks\nfJ89exYHDhxg13sIe+ONN7B27VosWbIEd955J5YvX45Vq1Zxw+UQVVpaim+//RYdHR04c+YMjh07\nBsdxMGHChKBLoz4oKioCgBs2/rS0tKQfI6JguK6LsrIy1NfX45tvvuGSkyHMtm2UlpZixowZ2L17\nN7Kzs7F3795uXxNY+N67dy9M00RZWVlQJVA/CSEgy9f+CMmyDMGLpg5plmWhsLAQFy9eRFVVFR57\n7LGgS6I+GD9+PIqKilBVVZUei8fjOHLkCO69994AKyMa2ZLJJJYuXYr6+nocOnQIBQUFQZdEA8jz\nPPi+3+1zAlnzLYTA7t278eSTT8K27SBKoAGwePFivPXWWxg/fjymTJmCuro6vP3223jmmWeCLo36\noKqqCp7nYdKkSfjtt9/w6quvYvLkyVixYkXQpdEtdHR04OTJkwAA3/fR2NiI48ePIz8/H2PHjsXK\nlSuxefNmTJo0CRMnTsSmTZsQiUSwbNmygCun6/U0lxcvXkRjYyNaW1sBACdPnkRWVhaKi4tvWNdP\nwepuLktKSvDEE0+gtrYWX3zxBYQQ6f0ZOTk5ME0zyNLpOt3NZU5ODrZs2YJFixahqKgI586dw44d\nO9DU1IQlS5Z0/8YDdAaWXjl48KCQZVnU1NQE8e1pgLS3t4vVq1eLcePGCcuyRGlpqXj99ddFIpEI\nujTqg08//VRMmDBBGIYhiouLxUsvvSQuXboUdFnUjUOHDglJkoQkSUKW5fTXK1asSD9nw4YNori4\nWJimKWbNmiV+/vnnACumW+lpLisrK2/6+MaNGwOunK7X3VyeOnXqhvGuW0+nsaPM624uo9GoePzx\nx0VJSYkwDEOUlJSIxYsX/6NsKwnBNQJERERERJkQ6NlOiIiIiIhGEoZvIiIiIqIMYfgmIiIiIsoQ\nhm8iIiIiogxh+CYiIiIiyhCGbyIiIiKiDGH4JiIiIiLKEIZvIiIiIqIMYfgmIiIiIsqQ/wcvOtu3\nN9Im/AAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import nonlinear_internal\n",
"nonlinear_internal.plot1()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So what happens when we try to linearize this problem? The radar gives us a range to the aircraft. Suppose the radar is directly under the aircraft (x=10) and the next measurement states that the aircraft is 3 miles away (y=3). The positions that could match that measurement form a circle with radius 3 miles, like so."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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H/IR9LlmpPwmdmD2PsKA9MuKcrhtEU/kTs+RDaQ4NJNnfk6Cgls/q/scK5EGvk8WdES7p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\nivz/vVagAotwNFgAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"nonlinear_internal.plot2()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see by inspection that the probable position of the aircraft is somewhere near x=11.4, y=2.7 because that is where the covariance ellipse and range measurement overlap. But the range measurement is nonlinear so we have to linearize it. We haven't covered this material yet, but the EKF will linearize at the last position of the aircraft - (10,2). At x=10 the range measurement has y=3, and so we linearize at that point."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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uWVDHDYsa8HusFRqxuFiiiTxa0UpDTZ1M/owxCd9CjJOjkRRP/X4rb79/CFUb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\nDUlEQVSbiIiIiGiK/P/HD9q87H8HHgAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"nonlinear_internal.plot3()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we have a linear representation of the problem (literally a straight line) which we can solve. Unfortunately you can see that the intersection of the line and the covariance ellipse is a long way from the actual aircraft position."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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e4umT70K5ZnEDn7x6KUtaQzM4OjFf6bpOPJ0jmswSSykkcqmxXtxWK/jcFmpC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\nQAAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"nonlinear_internal.plot4()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"That sort of error often leads to disastrous results. The error in this estimate is large. But in the next innovation of the filter that very bad estimate will be used to linearize the next radar measurement, so the next estimate is likely to be markedly worse than this one. After only a few iterations the Kalman filter will diverge, and start producing results that have no correspondence to reality.\n",
"\n",
"Of course that is a rather large covariance ellipse for an aircraft - it spans miles. I exaggerated the size to illustrate the difficulties of highly nonlinear systems. In real radar tracking problems the nonlinearity is usually not that bad, but the errors can still accumulate. And other systems you may be working with do have this amount of nonlinearity - this was not an exaggeration to make a point. You will always be battling divergence when working with nonlinear systems."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The Algorithms"
]
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"You may be impatient to solve a specific problem, and wondering which filter to use. So let me quickly run through the options. The subsequent chapters are somewhat independent of each other, and you can fruitfully skip around, though I recommend reading linearly if you truly want to master all of the material. \n",
"\n",
"The workhorses of nonlinear filters are the *linearized Kalman filter* and *extended Kalman filter*. These two techniques were invented shortly after Kalman published his paper and they have been the main techniques used since then. The flight software in airplanes, the GPS in your car or phone almost certainly use one of these techniques. \n",
"\n",
"However, these techniques are extremely demanding. The EKF linearizes the differential equations at one point, which requires you to find a solution to a matrix of partial derivatives (a Jacobian). This can be difficult or impossible to do analytically. If impossible, you have to use numerical techniques to find the Jacobian, but this is expensive computationally and introduces error into the system. Finally, if the problem is quite nonlinear the linearization leads to a lot of error being introduced in each step, and the filters frequently diverge. You can not throw some equations into some arbitrary solver and expect to to get good results. It's a difficult field for professionals. I note that most Kalman filtering textbooks merely gloss over the extended Kalman filter (EKF) despite it being the most frequently used technique in real world applications. \n",
"\n",
"Recently the field has been changing in exciting ways. First, computing power has grown to the point that we can use techniques that were once beyond the ability of a supercomputer. These use *Monte Carlo* techniques - the computer generates thousands to tens of thousands of random points and tests all of them against the measurements. It then probabilistically kills or duplicates points based on how well they match the measurements. A point far away from the measurement is unlikely to be retained, whereas a point very close is quite likely to be retained. After a few iterations there is a clump of particles closely tracking your object, and a sparse cloud of points where there is no object.\n",
"\n",
"This has two benefits. First, there is no process model. You do not need to predict your system state in the next time step. This is of huge value if you do not have a good model for the behavior of the system. For example, it is hard to predict the behavior of a person in a crowd for very long. You don't know where they are going, and they are endlessly being diverted and jostled by the crowd. Second, the algorithm can track arbitrarily many objects at once - some particles will match the behavior on one object, and other particles will match other objects. So this technique is often used to track automobile traffic, people in crowds, and so on. \n",
"\n",
"The costs should be clear. It is computationally expensive to test tens of thousands of points for every step in the filter. But modern CPUs are very fast, and this is a perfect problem for GPUs because the algorithm is highly parallelizable. If we do have the ability to use a process model that is important information that is being ignored by the filter - you rarely get better results by ignoring information. Of course, there are hybrid filters that do use a process model. Then, the answer is not mathematical. With a Kalman filter my covariance matrix gives me important information about the amount of error in the estimate. The particle filter does not give me a rigorous way to compute this. Finally, the output of the filter is a cloud of points; I then have to figure out how to interpret it. Usually you will be doing something like taking the mean of the points, but this is a difficult problem. There are still many points that do not 'belong' to a tracked object, so you first have to run some sort of clustering algorithm to first find the points that seem to be tracking an object, and then you need another algorithm to produce an state estimate from those points. None of this is intractable, but it is all quite computationally expensive. \n",
"\n",
"\n",
"Finally, we have a new algorithm called the *unscented Kalman filter* (UKF) that does not require you to find analytic solutions to nonlinear equations, and yet almost always performs better than the EKF. It does especially well with highly nonlinear problems - problems where the EKF has significant difficulties. Designing the filter is extremely easy. Some will say the jury is still out, but to my mind the UKF is superior in almost every way to the EKF, and should be the starting point for any implementation, especially if you are not a Kalman filter professional with a graduate degree in the relevant mathematical techniques. The main downside is that the UKF can be a few times slower than the EKF, but this really depends on whether the EKF solves the Jacobian analytically or numerically. If numerically the UKF is almost certainly faster. Finally, it has not been proven (and probably it cannot be proven) that the UKF always yields more accurate results than the EKF. In practice it almost always does, often significantly so. It is very easy to understand and implement, and I strongly suggest this technique as your starting point. "
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"## Summary"
]
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"The world is nonlinear, but we only really know how to solve linear problems. This introduces significant difficulties for Kalman filters. We've looked at how nonlinearity affects filtering in 3 different but equivalent ways, and I've given you a brief summary of the major appoaches: the linearized Kalman filter, the extended Kalman filter, the Unscented Kalman filter, and the particle filter. \n",
"\n",
"Until recently the linearized Kalman filter and EKF have been the standard way to solve these problems. They are very difficult to understand and use, and they are also potentially very unstable. \n",
"\n",
"Recent developments have offered what are to my mind superior approaches. The UKF dispenses with the need to find solutions to partial differential equations, yet it is also usually more accurate than the EKF. It is easy to use and understand. I can get a basic UKF going in a few minutes by using FilterPy. The particle filter dispenses with mathimatical modeling completely in favor of a Monte Carlo technique of generating a random cloud of thousands of points. It runs slowly, but it can solve otherwise intractable problems with relative ease.\n",
"\n",
"I get more email about the EKF than anything else; I suspect that this is because most treatments in books, papers, and on the internet use the EKF. If your interest is in mastering the field of course you will want to learn about the EKF. But if you are just trying to get good results I point you to the UKF and particle filter first. They are much easier to implement, understand, and use, and they are typically far more stable than the EKF. \n",
"\n",
"Some will quibble with that advice. A lot of recent publications are devoted to a comparison of the EKF, UKF, and perhaps a few other choices for a given problem. Do you not need to perform a similar comparison for your problem? If you are sending a rocket to Mars, then of course you do. You will be balancing issues such as accuracy, round off errors, divergence, mathematical proof of correctness, and the computational effort required. I can't imagine not knowing the EKF intimately. \n",
"\n",
"On the other hand the UKF works spectacularly! I use it at work for real world applications. I mostly haven't even tried to implement an EKF for these applications because I can verify that the UKF is working fine. Is it possible that I might eke out another 0.2% of performance from the EKF in certain situations? Sure! Do I care? No! I completely understand the UKF implementation, it is easy to test and verify, I can pass the code to others and be confident that they can understand and modify it, and I am not a masochist that wants to battle difficult equations when I already have a working solution. If the UKF or particle filters start to perform poorly for some problem then I will turn other techniques, but not before then. And realistically, the UKF usually provides substantially better performance than the EKF over a wide range of problems and conditions. If \"really good\" is good enough I'm going to spend my time working on other problems. \n",
"\n",
"I'm belaboring this point because in most textbooks the EKF is given center stage, and the UKF is either not mentioned at all or just given a 2 page gloss that leaves you completely unprepared to use the filter. This is not due to ignorance on the writer's part. The UKF is still relatively new, and it takes time to write new editions of books. At the time many books were written the UKF was either not discovered yet, or it was just an unproven but promising curiosity. But I am writing this now, the UKF has had enormous success, and it needs to be in your toolkit. That is what I will spend most of my effort trying to teach you. "
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"## References\n",
"\n",
"[1] https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb"
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