{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Nonlinear Filtering" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<style>\n", "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n", "@import url('http://fonts.googleapis.com/css?family=Lora');\n", "\n", ".CodeMirror pre {\n", " font-family: 'Source Code Pro', Consolas, monocco, monospace;\n", "}\n", " div.cell{\n", " //width: 950px;\n", " margin-left: 0% !important;\n", " margin-right: auto;\n", " }\n", " div.text_cell_render{\n", " font-family: 'Lora';\n", " line-height: 125%;\n", " font-size: 100%;\n", " text-align: justify;\n", " text-justify:inter-word;\n", " }\n", " div.text_cell code {\n", " background: transparent;\n", " color: 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" }\n", "</style>\n", "<script>\n", " MathJax.Hub.Config({\n", " TeX: {\n", " extensions: [\"AMSmath.js\"],\n", " equationNumbers: { autoNumber: \"AMS\", useLabelIds: true}\n", " },\n", " tex2jax: {\n", " inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n", " displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n", " },\n", " displayAlign: 'center', // Change this to 'center' to center equations.\n", " \"HTML-CSS\": {\n", " scale:95,\n", " availableFonts: [],\n", " preferredFont:null,\n", " webFont: \"TeX\",\n", " styles: {'.MathJax_Display': {\"margin\": 4}}\n", " }\n", " });\n", "</script>\n" ], "text/plain": [ "<IPython.core.display.HTML object>" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#format the book\n", "%matplotlib inline\n", "from __future__ import division, print_function\n", "from book_format import load_style\n", "load_style()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introduction\n", "\n", "The Kalman filter that we have developed uses linear equations, 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. \n", "\n", "There can be nonlinearity in the process model. Suppose we want to track an object falling through the atmosphere. The acceleration of the object depends on the drag it encounters. Drag depends on air density, and the air density decreases with altitude. In one dimension this can be modelled with the nonlinear differential equation\n", "\n", "$$\\ddot x = \\frac{0.0034ge^{-x/22000}\\dot x^2}{2\\beta} - g$$\n", "\n", "A second source of nonlinearity comes from the measurements. 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", "$$x=\\sqrt{\\mathtt{slant}^2 - \\mathtt{altitude}^2}$$\n", "\n", "These facts were not lost on the early adopters of the Kalman filter. Soon after Dr. Kalman published his paper people began working on how to extend the Kalman filter for nonlinear problems. \n", "\n", "It is almost true to state that the only equation anyone knows how to solve is $\\mathbf{Ax}=\\mathbf{b}$. We only really know how to do linear algebra. I can give you any linear set of equations and you can either solve it or prove that it has no solution. \n", "\n", "Anyone with formal education in math or physics has spent years learning various analytic ways to solve integrals, differential equations and so on. Yet even trivial physical systems produce equations that cannot be solved analytically. I can take an equation that you are able to integrate, insert a $\\log$ term, and render it insolvable. This leads to jokes about physicists stating \"assume a spherical cow on a frictionless surface in a vacuum...\". Without making extreme simplifications most physical problems do not have analytic solutions.\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 compute an approximate solution. \n", "\n", "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 teach you how to design and implement different kinds of nonlinear filters." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Problem with Nonlinearity\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 them we get another Gaussian as a result. That is very rare. $\\sin{x}*\\sin{y}$ does not yield a $\\sin$ as an output.\n", "\n", "What I mean by linearity may be obvious, but there are some subtleties. The mathematical requirements are twofold:\n", "\n", "* additivity: $f(x+y) = f(x) + f(y)$\n", "* homogeneity: $f(ax) = af(x)$\n", "\n", "\n", "This leads us to say that 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 such as a hum for a zero input the additive relationship no longer holds. This is because you linearity requires that $amp(voice) = amp(voice + 0)$ This clearly should give the same output, but if amp(0) is nonzero, then\n", "\n", "$$\n", "\\begin{aligned}\n", "amp(voice) &= amp(voice + 0) \\\\\n", "&= amp(voice) + amp(0) \\\\\n", "&= amp(voice) + non\\_zero\\_value\n", "\\end{aligned}\n", "$$\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!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## An Intuitive Look at the Problem\n", "\n", "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. The reported distance is 50 km, and the reported angle is 90$^\\circ$. 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 gain insight, so let's see how it fares for this problem. 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 the answer will be x=0 km, y=50 km.\n", "\n", "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." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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C2dlZ9O3bV/Tu3Vv01+mEM4iZXbqI/fv3N3cTrxv1+burJCcnX/YBJiIiQiQk\nJGifn332WYvtd9xxx2X3r5ncSH3QVPu0wWAQQjS9u5C8DzQOUphLGkuYX1Ucc4lEIqlNbm4u3t7e\nZGVlafF2c3Nz8fPzA0AIAUB4eDiLFy8mPDycJUuWkJSURHp6OgkJCVpdffv21ZJHANxzzz3odDqK\nioq0hGWdOnUCYPPmzRbJeKytrdHpdPj7+7Nlyxbc3Nw4deqUVn9oaChr164lNzeXpKSky55Tx44d\n2bBhg8W65o4n3JgMGzZMS5x08OBBlFOncJw+Hd2PP4JOB926NXMLrx92dnZaspTatG7dGoCIiAis\nrKwYOnQohYWFxMfHAzB69GiL8ufOnbP43KFDB3744Qfts6+vLxkZGQD079/fIvb9kSNHALR4+iaT\nCZPJhIeHB2lpaZSXl5OZmYmnp6eMfy6R3MQ0OPOnRCK5tVGFg9ls1rIgurm5MXv2bMLDwwkPD0dR\nFHbt2gXAf/7zH4QQLF++nISEBPR6PYqiMG7cOE6ePMmYMWNwcHAA4JtvvtESWfTv3x+AvXv3alkm\nN2/eTHBwMIBFhk6oFtKrVq0iMjKSyMhIVq1ahU6nw8PDg/T0dFJSUsjIyMDT07PpL1ILp8vWrdj8\n9FP1h+JiWL++eRvURFwqc+yl8PT0ZMuWLYwePZq4uDjatWvHmDFjADAYDHh6euLp6UlISAiAltBJ\nr9cTGRnJV199hV6v1/pzSUmJVvfXX3/NmTNnsLa2BqoTSG3YsIFZs2YBEBcXx7hx43jooYe4cOEC\ngYGB+Pn5kZOT0+DzkkgkLZimMehfHdKVpXmQrzAlDekDNV07aiaQUSdOUsPFRA1ZFxwcrCVkURcH\nBwcxadIksW7dOi2qhqIoFsmAAgICNBcVdSJozTjfNV1MWlIouhuBAwcOiIv33CME/LHcd19zN6tJ\nuNa+8WcTTlV3mJKSEi3GujpnombIR3Uy6ahRo+p1bQkJCdFctmouY8eOtZgsWrsdVxI7/lLI+0Dj\nIF1ZJI3lyqII8b/3zM1IVVVVnZTK7du3x8pKGvSbkoMHD2IymdDpdAwdOrS5myNpBhrSB7Zu3Yq3\ntzebNm1i06ZNJCYmAtUp2Tdv3mxRVq/Xa+nZBw8ezGOPPQb8kbZeZcuWLXh4eBAbG4vRaGTcuHHk\n5+ejKArp6elaOXd3d/Ly8upNay5TntfDr7/CvfdCZSXY2lps+r2iAqv//hddaekfK7t1gw4d6tZT\nXg7W1nDoUP3bWzjXo2+oxzCbzfj7+5OVlYXZbMbX15fIyEgAYmNjtfLOzs7ccccdWFtbs3nzZuq7\nJa9evZpJkybh4+NDfn4+BoNpPpYLAAAgAElEQVSBIUOGMHnyZAIDAwkODqZ169YWv4v6XMzqQ94H\nGodTp07Ru3fvpqn89dfh1Vebpm6grKwMIQSKomBnZ9dkx7nZqa8PXIu+lcL8FkYOyJLafeBKhIta\nZuTIkbz88sv06dOHuLg4HB0dKSoqYuTIkXz66ae0b9++zu86OTlZE/K9evXi+++/17aNGTOGjz/+\nGICcnBzKy8v57LPPABg5ciTe3t5SaDeEL76AwED49lswm69+fxsbsLeHlJRqkS+5LDV/S4D2f1pa\nGpMnTwYgJCREm+8wd+5cYmJiAOjVqxcBAQG8+eabQPWDLUBBQQEAiqLg7OzM7t27tc/Z2dkAmhhX\nxfmfPYTI+0Dj0GTC/D//gf794Ztvqn9/TYAU5o1DYwlzqXwlEomGamXLzc21WG8ymcjMzCQzM5Os\nrCy8vLx48cUXSUhIwGw2YzQaefjhhwHo3r07gMVgNHLkSKB6gmbfvn0B6ojy7OxsdDodubm5+Pv7\nk5GRweLFi2nTpg0BAQFSlDeU++6DAwfAzw86d766fW+7Dfz94fPPpSi/QtTJwjqdzuL/w4cPAxAU\nFIS9vT2JiYmkp6cTERGh+ap///33nD17luDgYO1tkyrKoXpCtSrK27dvz/r163Fzc6O8vJzXXnuN\ngoICTCaTdkyw/A1LX/QbiH/9q/rv2rXN2gzJ9UMKc4lEAlTfuM1mM2lpaXh4eFjcyNPS0vD19cXX\n15c9e/YghKC8vByoFtuRkZGcPXsWwCLSRJcuXQA4e/YsiqJgY2OjRbJQFIWxY8cC8Morr2hiws3N\njaioKN544w2MRiNhYWHX8zLc3NjZwaZN8MYb0LPnle3Tqxe8+SYkJ1fvfwvSmJMrIyIiMBqN9O/f\nn9jYWIqKivDx8cHOzo7Vq1cza9Ys9Ho9FRUVJCcnM3z4cAsrnKOjo/a/lZUV//3vf0lLSyMvL49J\nkyYRFxdHbGwso0aNIiEhQRPiaqQkPz+/Og/ekhZMWlr1zI9a7oHXi7Vr16IoCoqikJeXV2e7EIJ7\n7rkHRVFwc3O77u27GZHhEiUSCSaTiZiYGGJiYrRwbqqft6IoWlhCPz8/zVL3zTffANUC22g0kpyc\nzNixY9m5cyejR4/WIrH06dOH7777DldXV9zd3dHpdNoxVJ9Ys9msvYIHiI6OZsSIEZcMYydpII8/\nDuPGwcMPw7Fjly7n4ADbtkGfPtevbS2QK/XXvhLU8IxlZWW0atXK4sHTzs6OBx98kKVLl2rrCgsL\nOXXqlPa5qKgIPz8/du3aRXFxMQDbt2/H19cXBwcHjh07hpWVFYcPH9bmcmRkZODu7s7cuXMZMmSI\nFkFJp9NhMpkoKCjQ3GXkHI0WxHffgRot6fx5OHUKmsqP/U9o3749q1evriO+d+zYwX/+8x/at2/f\nLO26GZHCXCKRsGfPHs2/dffu3fz+++/Mnz+fV199lTNnzlBWVgbAvn37+P777+nTpw/Hjh3D1dWV\nxYsXaxPWevToodWhEhcXR3Z2NgkJCeTl5TFhwgR8fHy07RMmTMBkMml+sYDF/5Imok+fP5/A2aHD\nLS/KoTokaGP3yUvFT/fw8MBgMGi/x//7v//D1dWVyspKTCYTr7/+OlZWVhQXFzNgwADOnj3Lr7/+\nypNPPomvry/Hjh2jqqoKqI6zvmfPHgoLCwFYuHAhc+bMYcGCBaSkpGBra8vXX3/Nq6++yuuvv05x\ncbHFpNWbJW7/Dcv69dXhS+GPMKZz5jRLUyZPnsyGDRt466236FBj3Fi9ejWjR4/m119/bZZ23YxI\nVxaJ5Cbkal+9Ozs7a/6tcXFxpKSkIIRg3759JCYmapFTVL/w7777joCAAHbs2MGsWbMIDw8nODiY\nlJQUIiMjiYiI0GKNt23blrVr12oRV+rjUv64kiaktBTOnLl8mTNn4Jdfrk97WjDXs0/qdDoMBgMZ\nGRlkZGQQGBiIXq8nPj6etm3boigKAwYMQFEUAgMD+ctf/gJUR0Oy/9/kwJEjRxIcHMzu3bvp06cP\nMTExrFu3jsDAQAYOHAjAxo0b8fLyoqKigtdff52KigomTpzIhQsXiIqKws3NTcZHb26Sk+F/D1lU\nVVV/biamTJkCVPcblV9++YXU1FSeeuqpOuUrKipYsGABAwcOpHXr1nTr1o3p06dbxPIHSEpK4uGH\nH6ZHjx60adMGR0dHIiIiuHjxokW5adOm0a5dO7755hsmTJhAu3bt6N27NzNnzuT3339vgjNuPqTF\nXCK5Can96r3262n1c+fOndm7dy+jRo0iKCgIk8lEamoq3t7eDBo0iOnTp+Pp6aklBwK01+VqlsO9\ne/eSn58PVGfXnDNnDtu3b2fx4sVaEpabKWvmTcPmzVDjJlnZti3WHTvChQvV4RUBfvyxutz06c3U\nyFsTnU5n8VZJtdi7ubnh7OyMm5sbtra2zJ8/n5SUFNq2bYuTk5P2pqpPnz6kpKQwZMgQbbKpGsJU\nCMFrr73G4sWLATh27BgODg5auNOsrCySk5O5//77+fzzz4mOjpbW86biMmFMte01OXsW7rmnbrnr\nEMa0Q4cOBAYGsmbNGp577jmgWqRbWVkxefJkVqxYoZWtqqrC19eXnTt38tprr+Hi4sLJkyeZN28e\nbm5uFBYW0qZNG6A60daECROYMWMGbdu25ejRoyxevJhPP/1Ui9KlYjKZ8PHx4emnn2bmzJnk5+cT\nExNDx44diYqKapLzbhYaMbb6NSMTDDUPMrHEzUvt5CI1E5BUVFRoyUqeffZZYWVlJZ599lmhKIqW\nGCggIEAEBASIgQMH1psIJTAwUACid+/eWsIfg8GgHSMjI0MoiiIyMjKa+UpILomLi5ZQqKJLF3H6\n1VeFqKoSIj5eiNtv/yPZkF7f3C2V1EPtBF+KooiAgAABiISEBJGamirCwsKEXq8XgBg0aJD2+1V/\n546OjsLJyUlb7+rqKkpKSoTRaNSSIqkJjSSX55oTDB08KET//kLY2Fgm+brSxcZGiAEDhPjii2tu\n++USDP3rX/8SgNi3b5/45JNPBCAOHz4shBBixIgRYtq0aUIIIQYPHixcXV2FEEJs3LhRACI1NdWi\nrn379glAvP322/W2o6qqSphMJrFjxw4BiIMHD2rbnnzySQGI5ORki30mTJggHBwcrvncG5PGSjAk\nXVkkkpuQ2q/ea/rI5uTksGDBAqDa+h0fH6/5pAYGBhIaGsrmzZvZvHkzR48erVN3ZGSkFoP11KlT\nCCEYNWoUBoNBO4anpyfZ2dky3X1L5cKF6ollNjaU9+3Lsbfe4sepU0FRYNYs2L4dBgwAnQ5OnoRa\nr5Ul15/abiX1udds3rwZRVFo3bo1qampxMfH4+rqitFoZPv27VoUpEmTJhEcHExRUREHDhygY8eO\njB8/npSUFN566y3mz5/P4cOHURSF4cOHX7ULj3SBuQpuoDCmrq6u3H333axZs4ZDhw6xb9++et1Y\nsrOz6dSpE97e3pjNZm1xcnLijjvusIjucvz4cUJDQ7UkWzqdDldXV6B6onNNFEXB29vbYt19993H\nyZMnG/9kmxHpyiKR3MBcSQSFsrIyLQGQ0WiksrISIQTh4eEUFRVhNptZu3YtY8eORQjBqlWr6NWr\nF/Hx8dxzzz188803vPTSS3To0IFhw4bh7e2tTQbt06cPo0ePrtddRb76bsFkZVW7qUyaxFevvEKF\ntTUWvWfIkGqxMG0aZGZWl//fHARJ83C5yDCenp6kp6djNpuxsam+rW/cuJGQkBAqKysxGo0cOXKE\nnTt3YjAY8Pf311xcoNpX+KOPPuKvf/0riYmJREZG4uTkREpKCmazmczMTO03filqjkWNGcXmlkAN\nY/r++xAe/udzP6A6jOnixfC/yDvXA0VRmD59On//+98pLy9nwIAB2sNeTc6dO0dpaSmtWrWqtx51\nztKFCxcYO3Ystra2LFiwgAEDBmBnZ8epU6cICAjgt99+s9jPzs4O21ouP61bt9ZC994sSGEukdzA\n1HcDrJ1x8Omnn9b8R+GP8Ia//fYb7777rrZ+586d7Ny5k4yMDG3wa926NQB5eXl8+eWXzJs3D29v\nbzp27KhlLJTcgJSUVCcsmTwZcfAg1GfZbNMGkpIgMfGPyBCSZuNykWFq+6SbTCays7Mxm834+fkB\n1Vl3Ac0CHhERwbFjx7T148eP18aJ/Px8YmNjCQkJ0dYNGTKEbdu28fnnn+Pm5kZeXp6FQaDmWNQU\nUWxuCW6AMKbTpk0jKiqKd955h4ULF9ZbpmvXrnTp0oUPPvig3u1qaMWPP/6YM2fOkJeXp1nJAUpL\nSxu/4TcQUphLJDcw9d0Ac3Jy8PX1JSIighMnTmg3Vn9/f/r168f333/PjBkziI6O1vZ56KGHGDZs\nGDqdDrPZzMKFCzEajTz77LNMmTKFjRs3snLlSi2+uLSC3eD87W9XXlZaylsEVzOBWi1rMpk0SzqA\njY0Nnp6emEwm8vLymDx5sibM+/btS1BQEJs2beLf//4348aN0x6+e/fuzeHDh/Hw8ODIkSPMnTuX\nhQsXWhgEao5FcrJ3A2jhYUx79epFWFgYR48e5cknn6y3jJeXF4mJiVRWVjJq1KhL1qUoCvCHAUjl\nn//8Z+M1+AZECnOJ5AalPjcWk8mkxSyOi4sD0BL8pKWlERwcTHJyMg4ODpw+fVqr6+OPP2bcuHFE\nR0ezadMmoqKiCAsLw87OTvMHjIyMZMSIEdIKJpHcINS2pKts3boVLy8v5s6dy6JFi0hKSmLNmjUW\nZZ5++mkqKyspKCjAy8uLnTt34u7uzpEjR7j33nu1kIpw6bFIJiq6Bq4mjGnHjtenTbVQ7y2XIiQk\nhA0bNjBhwgRefvllRo4ciU6n4/Tp03zyySf4+vri7++Pi4sLt912G88//zzz5s1Dp9OxYcMGDh48\neJ3OpGUiJ39KJDcQ6qSqsrIyYmJimDhxIlOnTtV8vrOysoiJiSE8PJzk5GTGjBnDd999x+DBgwG0\nSZ5OTk6sXLmSBx54AIBnnnmGsLAwsrKysLGxITo6mu3bt5OZmaml9JbxxSWSmwMPDw/mzZvHwoUL\nyc/P58CBA9q2oKAgFEWhqKiIgoICoNon+PDhw6xYsYKAgABSUlKYP38+8fHxmgD39vYmNzdXq6e+\ndZIroFYYUzp0gJ49La3oahjTFoq1tTWZmZlERkayefNm/P398fPzIy4uDltbW+7930TVLl26sGXL\nFuzs7Hj88cd56qmnaNeunXSTbJKYMVeJDJfYPMhwiS2b2iEPhfgj7KHRaBSKomhhzoxGo6ioqNDC\nGM6aNUtkZGSI8PBwrYxerxdWVlYiNDRUC4u1b98+sWLFCrFv3746x1VDsCmKIrZs2XLdz19yfZDj\nwK3JxYsXhdFoFJ988ol45plnxJIlS0RGRoZITU0ViqJoY0dwcLBYvXp1nZCpY8aMEYqiiOTkZGEw\nGERqaqq4ePGiNmbVN37dzFxzuMTa1AhjKm6/XYilS5s8jOnlwiVKrpzGCpcoXVkkkhZKfRM73dzc\nmDNnDo6OjqSkpFBeXk5GRgaOjo6kp6eTkpICwK5du1i2bBlCCAYPHsyXX35JQUEBoaGhbNy4kcce\ne0yzfuv1egsreG3/VEC6r0gkNxl5eXlER0fz9NNP4+joiIuLCz/99BPu7u5kZ2dTXl6Ooig89thj\nWrbH3r17c+rUKQC6d++OEIKUlBSSk5MJDQ3FxsYGf39/Nm3axKFDh5gxYwa5ubn1ThaV1EONMKbY\n21dbxf/3tpNZs+DRR2HSJPj22z/CmLZt27xtljQ61kaj0djcjRBCUFFRYbGudevW2sQASdNw7tw5\nqqqqsLa25o477mju5khq0a9fP0aNGoWHhwfW1tZAtVh//vnnSUlJYcCAAfTr14+oqCg2bdrEnj17\n+PV/meLUmyfA0qVL+emnn1i0aBFhYWG4uLhodV6uD1hbW+Pg4ICDg4N2fMnNhxwHbk369euHlZUV\nr7/+Orm5udjY2DBr1ixcXFyYMGECffv2RafTERISgqIonDt3jvHjxzNu3Dh69uxJamoqiqIwZ84c\nfvzxR7Zu3UpISAiPP/44e/fuJSYmhhMnTjB37lx0Oh3PP/88o0aNon///s196k3Cr7/+SseG+nyn\npsLGjdWxyT/4oNqFpSbdu8Mzz1RHbPn8c3B0rA5t2kDUePOKosgHpwZQXx+4Fn0rLeYSSQukdsjD\nrVu34ubmRnl5OUFBQSQnJxMTE0NkZCQuLi4UFBRw6tQpzaIVFhaGTqdj2LBhQHUoxJkzZ2JnZyej\nJUgkEnQ6HZGRkXTq1InKykpcXFzw9vbWxhzVog6g2u9Un3MVNRdCfn4+c+fO5ffff6ewsJA777wT\ngHHjxtG/f38cHR1JS0vDw8NDTgq9HDXCmF4SGcb0pkcKc4mkBWEymcjJyWHv3r0sWrSIxMRE0tLS\nSExMRK/XU1BQgKIoREZGYm1tTUxMjMX+QUFBLF++HJ1OR1xcHFlZWVqoNIlEIqmJTqfDzc3NYnK3\nihr+0M3NjcrKShYsWEBQUBCBgYEcOHCA2NhYrKystMnmX3/9tTYe9flfKL+33nqLI0eOAGhucbGx\nsURHR8vkQ/Uhw5hKkMJcImlR5OTk4OfnhxACRVF46623yM/PB/6wVgkhGDp0KLa2tlr4QxcXF+68\n806GDh2qba8Z3zw7O1v6iUskkiumplA3GAw4OztrVm4/Pz/0er3mh37y5EkSExMZO3YsY8eOpX//\n/kyfPp1HH30ULy8vFi9ejNlsJjc3l/nz5zNv3jw5Hkkkl0AKc4mkmVFf7bq5uVFYWIgQwsIirtfr\n6dq1K4qiYGVlxebNmzl8+DALFixg9uzZFtn5rKysUBQFW1tbC2uUtExJJJJrpeaEcNWtDuDRRx8l\nOzsbvV4PQGJiIi+99BKFhYUMGjSIZcuWEVLDsms2m0lPT8fV1ZXY2FjN5U66tliyf/9+EhISKCgo\n4Oeff9YMNbfddht6vZ7Q0FDNTVFy8yGFuUTSDKguK2azmQMHDrBw4UKioqK0jJszZsxg2bJlhIeH\ns3jxYm0/NRufl5cX1tbWREdHk5aWRlBQEAcOHGDmzJk88cQT0holkUgaHTVSVFRUlDb22NjYYGdn\nx5QpU0hKSiI1NVXLJjp48GASExMxGAyYzWYmTZpEZGQkhYWFmtvLiBEj6kSfulVJTEwkJiaGc+fO\ncf78+XrL7N27l3Xr1nH77bdjMBgsHnwkNwdSmEskzUBubi5+fn5AtWuKwWAgLCyMESNGoNfr8fX1\nZceOHQQHB2v79OnTh/z8fO68806Cg4MtMnHqdDoCAgIAaR2XSCRNQ02/8xEjRmA2mzVR7enpSXZ2\nNhcuXCA5OZnBgwcTERHB1KlTGT58OHv37gWqfcyTkpIwGo2axTwtLQ2z2az5ut9qlJSUMGXKFPbv\n309paemflj9//jznz5/nhRdeYPXq1SQkJNCtW7fr0FLJ9UBm/pRIrjMmkwmz2UxKSgqzZ8/W1i1c\nuJDy8nKWLFnCjh07GDJkCL6+vkB1iuM9e/YQGhrK22+/DSAzcUokkuuKOuao0Z08PT21uSzqNn9/\nf4xGI0VFRdjZ2TF37lzMZrOWfRjg4MGD3H///UC1kcJsNuPn50dOTg7wR4ZjNYzfzcyJEydwdnbm\no48+uiJRXpPS0lK2b9+Os7MzJ0+ebKIWSq43UphLJE2IeoMpKyvTbjRpaWn4+vpiNpuZM2cOoaGh\nLF68mNjYWAIDA7V9v/zyS2xtbTEajaxevZoePXqwYcOGhsfKlUgkkkbgUsaB+++/X3NzWbBgAZMm\nTeLLL79EURQCAwOJjY3F19eXJ554gokTJ5KUlIQQAqgeM2NjY/Hy8iI3N7c5Tuu6UVJSwvjx4zl+\n/HiD6jl+/DgPPfQQP/74YyO1TNKcSFcWiaQJqc8n8x//+AcAmzZtoqioiISEBCIjIwEYNmwY48eP\nt6gjOjqaESNGSBcViUTS4snNzcXf31+zpM+dO5eYmBiGDRtGdna2FmUKYPPmzQAkJycTEhKCu7s7\nubm5REdHW0RuuVljn0+ZMqXBolzl+PHjTJky5aZ/mLkVkBZziaQJUX0yZ8yYQVRUFGazmZ07d2rb\no6OjCQ0NZc6cOSxcuBBvb2+WLVtGbGwsixYtwsbGxiLsoUQikbRk1DFPFdEGg4EtW7bg7e3NhAkT\neOCBBwBwdHTU5tlA9cTH7du3U15ezpw5c5gxYwa5ubnaRPmJEycSExNz07i3JCYmsn///kats7Cw\nUIvQ1VisXbsWRVEoLCy8qv3KysowGo3k5eU16Pj9+vVj2rRp2uczZ85gNBo5cOBAnbJGo/GmyBgv\nLeYSSROivurdvHkzRqORDRs2YDAYqKysJDY2lpCQEBISEpg8eTI+Pj7k5uayYMECAObOnYunp+dN\nZSGSSCQ3N7UTFdX+bGtrC8DRo0fx9/dn4MCBCCG477772LhxI4mJiSiKokWdioqK4t5770VRFGJi\nYnBycsLW1vaGt57HxMRctU/5n1FaWkpMTEyLiNRSVlbG/PnzAbTwmtdCWloaHTp00D6fOXOG+fPn\n069fP5ycnCzKPvPMMzz66KPXfKyWgrSYSyRNQM3JSyaTiU2bNgGQkZHBggULGDp0KEajkYCAAIsn\nfA8PD9LT08nIyMBgMNzQNx6JRCKpjaurKyEhIYSHh7No0SJ0Oh1LliwhKyuLxMREQkJCSElJYdCg\nQQQHB2M0GrGxsWHu3LkAHDhwAG9vb3Jycm7YCaL79+/n3LlzTVJ3cXExn332WZPU3Rzcf//93H33\n3VdU9s4778TZ2bmJW9T0SGEukTQBqm95Tk4OsbGxmg9lnz59EEJw8OBBoqOjad26NdnZ2Xh6egLV\n1iUfHx98fHykKJdIJDcdK1asIDExkePHjzN37lwcHR0B8PHxITIyEnt7eyorK5k8eTKJiYna/JuZ\nM2diNBqZOXMmWVlZlJeXM3HiRIxG4w0nzhMSEi4Zp7yh/PjjjyQkJDRJ3QDTpk2jXbt2fPPNN0yY\nMIF27drRu3dvZs6cye+//w5UR5pRwzfOnz8fRVFQFEVzSZk2bRr9+vWrU3d9rig1XVny8vIYMWIE\nANOnT9fqNRqNl9y/qqqKJUuWMHDgQFq3bk337t2ZOnUqp0+ftijn5ubGkCFD2LdvH2PHjsXOzg57\ne3vi4uKoqqpqyCW7aqQwl0gaGTUc4qZNmygsLMRoNBIVFUVQUBBLly4Fqid5qrF/ZchDiURyqxAW\nFkZoaCibNm1iwYIFFBUVAdC2bVtatWpFbGwsKSkpWpQWAH9/f1588UWMRiO+vr7o9XoOHz4MwKJF\ni264CY+7du1q0voLCgqatH6TyYSPjw/jx48nIyODp556iuXLl2vJ8Hr06MEHH3wAwNNPP83u3bvZ\nvXs3BoOhQcd94IEH+Ne//gVUu3qq9T7zzDOX3OeFF14gPDwcDw8PMjMziYmJ4YMPPsDFxaVOFJsf\nfviBxx57jMcff5zMzEw8PT2ZPXs277//foPafbVIH3OJ5CqpHSFAnZwE4O7uTlxcHAsWLGD27Nks\nWrQIg8FAZGQkOTk5KIrC3Llz8fb2lmJcIpHcctjZ2bFq1SruvvtunJycePTRR3F2dsbDwwN3d3cA\nBg0axKZNmxBCMGzYMObMmUNMTAx9+/Zlx44dvPjii6xatQoAJyenG25y/E8//XRD119RUcH8+fMJ\nCgoCYPz48RQWFpKQkEBUVBStW7dm2LBhQOO6l3To0IEhQ4YAcPfdd/9pvUePHmXlypW8+OKLvPnm\nm9r6+++/n1GjRrF8+XIWLlyorT9//jxbt25l5MiRQPX9PC8vj4SEBKZOndoo53AlSIu5RHKVqG4q\nasSA2NhY/Pz88PPzY+HChcTExFhYe4YPH45Op8Pd3Z158+YREREhRblEIrllycvLY+HChdja2mrJ\ninQ6HXZ2dsybNw8/Pz9SUlJ45ZVXWLFiheb2cPLkScaNG4evry86nY7o6Ggt4/GN5G9e8/5wI9av\nKAre3t4W6+67774Wl+Tok08+AbCI6gIwcuRIHB0d+eijjyzW33HHHZooV2mO85IWc4nkClEt5Xq9\nnqioKNzc3MjNzWX+/PmEh4dz/Phxjh07BlRn6pwzZw56vV6z5uTl5cmY5BKJ5JanZkjF2qjjrI2N\nDcuXLwfg559/RlEUgoKCuOeee5gyZQpRUVGEhYWRl5dHeXk5gYGBpKen4+Pjc71P56pp6pB+TV2/\nnZ2dFl1HpXXr1pSXlzfpca8W1Y+/R48edbb17NmzjuDu0qVLnXKtW7fmt99+a5oGXgIpzCWSK6R2\nsqARI0bg5ubGvHnz+OWXX0hOTgaqRfnq1asB2LdvH3q9noKCAtzc3GRMcolEcstTO4RiTdRxNi0t\njQ0bNvDPf/6T9957j9mzZ2uTGl1dXTEajXz11VckJiYSHBx8PZvfYDp37nxD199QbG1ttYmiNWns\nzKWq0D579ix33nmnxbYzZ87QtWvXRj1eYyGFuURyhahWHr1eD1TP4lat4KNHj9bKBQUFkZeXx549\ne4iJieGTTz4hPz+f7OxsaSmXSCSSy1A7QVFoaChbt25l48aNFjkggoODSUhIYNy4cSQmJhIaGqr5\nqLd0XFxc2LNnT5PVr96jmpPWrVsD1Gtt7tevH8XFxZw7d47bb78dqPZb//DDDxtUb20eeughAN5/\n/30tmgtUG8yKioqYM2fOn59IMyB9zCWSK8BkMpGVlcWePXvYtm0b8+fPZ/v27dpN5LnnngPQ/B29\nvb1xcnIiNDSUnTt3WqSXlkgkEkn9qNb0mvNwPDw8yM7OxmAwMGrUKBRF4Z577gEgPz+fcePGsXHj\nRv7yl79QVlYGWOaSaGmEhobW6zbRGHTt2pXQ0NAmqftqaN++PX379iUjI4Nt27ZRWFjIiRMnAJg8\neTLW1taEhISwdetWNmnHqBAAACAASURBVG/ezMMPP0xlZeWf1nv33XfTpk0bNmzYQF5eHoWFhZw5\nc6besg4ODjz77LO8+eabvPLKK2zbto2VK1fi5eVF7969eeWVVxrzlBsNKcwlkj/BZDIRExPDpEmT\niImJ4c0330QIgdlsJicnB7PZTNu2bQHYvHkzAFlZWXh7e7N27VqysrKIjIyUEz4lEonkGqgp1j09\nPcnOzmbOnDla+L2dO3cybtw4EhISiI+PByxzSdQU6CaTic2bNxMVFaWJ+OvNsGHDNEtxY9O9e3ce\neOCBJqn7alm9ejV2dnb4+PgwYsQILd74XXfdRUZGBqWlpQQGBhIWFkZQUNAVRT6xs7NjzZo1nD9/\nnocffpgRI0awcuXKS5b/f//v/xEXF8fWrVvx8vJizpw5PPzww+zatavJHo4aiiKaevruFVBVVcV/\n//tfi3Xt27fHyko+NzQlBw8exGQyodPpGDp0aHM3p8WydetWJk6cCMDYsWMpKCjQXoEtWLAAgPT0\ndAoLC1mwYMENMwEJZB+QyD5wq3Mjf/81Q9W6urqyYsUKwsLCsLOzo6ysjPj4eO69916CgoLIyspi\nwoQJZGZm4uvrC1QnpJk3b16jtOXUqVP07t37issnJibywgsvUFpa2ijHB+jUqRPvvPMOkydPvqr9\nysrKEEKgKAp2dnaN1p5bjfr6wLXoW6l8JZIa1Hz9aTKZyMzMpLy8nNTUVDIyMvjggw9IS0ujsrKS\nmJgYZs+eTUpKCgAREREWWTwlEolE0nSoFnSAHTt2EBkZqQnL7du3M3/+fPbv309aWprmSqhGDlEt\ntc1FSEgIw4cPb9Q6hw8fftWiXNLykMJcIqlBzRjlubm5+Pr6MmnSJGxsbPDx8UGn0/H555+zaNEi\noPqV5KFDh/Dz8yMvL09m8ZRIJJLrSG5urpZHorbbihCC2NhYbGxstHFZzRhaXFzc7H7oCQkJ2Nvb\nN0pd9vb2bNy4sVHqkjQvUphLbmlqD8w1IwJ4eHgQGRkJgNlsBtDils+dO5eMjAxsbGyIjo6Wkzsl\nEomkGfDw8CA9PZ309HSgeuJ9bGws7u7upKamYjAYcHNz08pHRETg6upKfn4+vr6+WvnmEOfdunXj\no48+arA4t7e35+OPP26x4f8kV0eDhfm7776Loii0a9euzrbPPvsMd3d32rVrR6dOnQgICOD48eMN\nPaRE0mBUQZ6Tk4OXlxexsbGUlZWRm5urJQ4CtLTCKjWjA/j4+ODp6Sknd0okEkkzodPp8PHxwcfH\nB3d3d0JCQrSoWYcOHWLhwoXk5eVp5e3s7MjIyMDV1ZUdO3Zo5dUx/3rTr18/9uzZw/jx4+nUqdNV\n7dupUyfc3d3Zu3cvffv2baIWSq43DRLm33//PbNmzaJnz551th09ehQ3NzcqKipITk5mzZo1fPXV\nV4wdO5aSkpKGHFYiaTCqywrAvHnziI6OZuHChUycOJHo6Gi8vLzIysriwIEDWha1zMxMcnJytPi6\n8P/bu/fgqMr7j+OfhF0Nlx+D4aKiolUoFGxAEzROBrvqbiWahCTghTqt4q0qKoyaipDbJiGVBqnQ\nqRegTi8IyMVFEhN0ERYdOtgBhKIdSwEtVqeCXBSBDLvk/P5gdptNNiGbbNizu+/XTEY852zyJPuc\nc777nO/zfUKX9gIAnHsej0fLli1TWVmZfD6fysvLNWvWLNlsNtXX1+vEiROqr6/Xe++9p/fff18l\nJSVatGiR6urqovrEc+DAgVq/fr1efvlljRw58qwj3wMGDNDIkSP1yiuvyO12M1IeZ7q0wNAjjzyi\nG2+8UampqYEJcH6lpaU6//zzVVdXp759+0o6M/o4bNgwzZ07V3PmzOnKjwa6xJ+y4n/E6XK59Ne/\n/lWS9K9//UuStG3bNv36179WWVmZLBaL8vPzJZ2pwGKxWIICdABAdPmfaDocDjU0NCgpKUkZGRny\neDzKzc3VrFmzVFVVpeeee07SmcmSVqs1UPrWbrfL4/F0+tp++vRp9ejRo9Ptv/vuu3X33Xdr27Zt\nWrZsmTZv3qzDhw8HKqakpqYqKytLP/vZz0xTEhFneL3eiFUS7HRgvmTJEm3atEn/+Mc/VFxcHLTP\n5/Oprq5Ov/jFLwJBuSRdfvnluummm+RyuQjMcc54vV653e5WI90Oh0PV1dVyOp2qq6sLpK34R8j3\n7NmjVatWBUbW/TmM0pk8Rn/5LQBA9PmfYEqS3W5XWVmZ7Ha7rFaramtrAxVZRo8eHdjnn+SflJSk\n4uJizZ49Wy6XK+zBl4EDB+rLL7/UJZdc0qXgXDoziNkyjRLmZBiGTp48qW+++UYDBw6MyPfsVGB+\n4MABTZ8+Xc8//7wuvfTSVvv37t2rkydPKi0trdW+tLQ0ud1uNTY2KiUlpc2f8cknn6ipqakzzUMH\nNV9wYefOnVFuTff54IMPNG3aNM2fP1/jxo0LbH/vvffkdDrlcDjUq1cvLViwQJK0evVqpaena8WK\nFTp27JiGDBkiq9UayOHzer2aP3++Bg0aFPN/t0TpA2gbfSCxxev7/8EHH6iiokL9+vVTjx49lJmZ\nKUlasGCB/vOf/8jpdKpfv37KysrSQw89pMWLF+urr75STU2N3n77bS1evFjz589XZmamtmzZoszM\nzLMG6YZh6Lvvvou5NViaL2fjH5hCxzQ1NQW+jhw50mp/cnKyhgwZEtb37FRg/thjj2n48OF69NFH\nQ+4/dOiQJCk1NbXVvtTUVBmGoSNHjujiiy9u82f4fL4OLc+KyDDjssWRMmbMGD344IO6+uqr5fF4\nlJ6erm3btunvf/+7DMPQu+++K0l699135XA4NH78eF177bWaM2eO1q1bp8suu0xTpkyRxfK/08V/\nkY+nv1s8/S7oHPpAYoun9z8jI0Pz5s3T6dOn9dRTT2nevHnKyspSZmam3n//fUlnUk98Pp9GjBih\nBx54QH/4wx/00EMPadGiRbr11ls1ZswYbd68Oej1Z+Ov4AVI6tTTk7AD89WrV6u2tlYfffTRWT9Z\ntbf/bK+1WCwx96kz1jS/CMdzrvSWLVu0ePFiJScna+HChXr44Ye1cOFCTZkyRZL0wAMPaPjw4XK7\n3Ro/frxuueUWSVJlZaWuuOIKLVy4UD/+8Y+DRtvjRaL0AbSNPpDY4vX9t1qtstlsgSecGRkZgZHv\nG2+8UQsWLAiMhhcVFWnevHlasGCBxowZo/3792vdunUaP368srKyAiPn8fT3aS5e+4AZdCaODSsw\n//777zV16lQ98cQTGjx4cGAp2VOnTkmSjh49KqvVqv79+0v638h5c4cPH1ZSUtJZywKNGjWKwLyb\nxfJSzOEYOXKkrrzySmVlZemiiy7S9OnTlZubK5vNpsLCwsBs/FGjRslmswVN/hk9erRyc3PjdqJn\novQBtI0+kNgS4f3PyMjQ2rVr9eSTT2rNmjXKy8sLrLo5evRoXXnllYFr/Nq1a/XOO+9Ikn7wgx8o\nIyMj4it0mk0i9IFoaWpq0rFjx8J6TViR7zfffKOvv/5aL7zwgi644ILA17Jly3T8+HFdcMEFuuee\ne3TVVVepZ8+e2rVrV6vvsWvXLg0dOrTd/HIgUppP/Ny8ebOcTqdefPHFQF65f6KQ/xj/7H1/TVvK\nIQJA/Go+YbS+vj6QilJSUiK73R7VlUGRmMIKzC+66CJt3Lix1dett96qlJQUbdy4UVVVVbJYLMrN\nzdWbb74Z9Elh//792rhxowoLCyP+iwChuN1u5eTk6L777lNWVlagZrnb7Q5aZMgfjDdf+RMAEB+y\ns7NVV1en7OzskPv994pt27apuLhYM2bMkMfjUU5OjiorK7V27VoCdJwTYaWypKSkBC1t6/fHP/5R\nPXr0CNrndDo1duxY5eTkaMaMGWpsbFRpaakGDBigp59+uqvtBjrE4XBo8uTJWrp0qX74wx+qqKhI\np0+fVmNjoxoaGlRQUCCXyxUIxpuPngAA4sPZru0Oh0NlZWVyOp2Szkzwt9lsmjx5sqqqqiT9r2Su\nz+eTxWJRdnY2T1MRcV1aYKg9I0aMkMfj0bPPPqtJkybJYrHo5ptv1ty5cyNW6xE4G6vVqkWLFgWC\nco/HE3SRbR6QAwASk9Vq1cyZM3XNNddIOhOou91uLVu2TMXFxYE88/z8/MCCP3V1dQzkIOKSjOYF\nLKMkVHL8//3f/zH5s5sl4oQPr9er2tpa7dixQzNmzFCvXr2i3aSoSsQ+gGD0gcTG+982/xwlf1EA\nm82m9evXtxoxD7WIXSyhD3SfzsS3RL6Iaf488Za5f21tt1qtSklJ0ezZs+XxeM5hSwEAZhfq3rF+\n/Xrl5ubK4/EoLy9PhYWFstvtqq6u1okTJ+R2u4OKBgBdQWCOmNbQ0KCcnBw1NDQEbW/vQskETwBA\nKM3vHf5/S2p1z6ipqVF5eblqamq4pyCiui3HHOguzR8btqXlhbLlo0byAgEALbW8d7Q1D2n69Ona\nvXu3pk+fHrin+EfbYzWlBebAiDliitfrVWVlZWCU3G63q6ysTHa7PbC/vr5ekoLqj/tLYVVXV1Py\nCgAQUvO1K1quY+H1erV27VqtXbtWmzZt0vLly7V58+bAa0lpQSQQmCOmuN1uVVZWyj9n2ePxqKKi\nQuvXrw/UJA8VgPtLYflrmAMAEA632638/Hzl5+fL5/OptLQ0qEw0KS2IBAJzxBSHw6G33npLb731\nlux2u3w+n1wulyQFcgH9AXjz4NxfCouLJgCgMxwOh9asWaM1a9bIYrGooqIiqIgAK0UjEsgxR0yx\nWq3Ky8uTdGb5ZP8CQZLkcrmUnZ0dWNnNv8iVP5+c3HIAQGc1v//4S+86HI5W855iuXQioo8Rc8Qs\n/2NDSSooKJDFYgnkBc6cOVN1dXWMjgMAOq290rv+RYgaGhpaVXIhZRKdRWCOmOUfAbfb7a1y/Xik\nCADoqvYC7eblFF0ul3w+n2w2W6uKYPX19Tpx4kTIAB9oicAcptaRi5p/AigLBgEAIqm9CZ3+fXa7\nXR999JEKCgoCK4M2NDToxIkTqq6uVm5urmpqaloF+G2NxiOxkWMO0/F6vUELBhUUFKi0tFQVFRWq\nra1tlSfOTHgAQHdob26Sf199fb2cTqfKysokSfn5+ZKkyZMna+nSpSovL1dRUZHGjh0bdJ/yj7iH\nuq8hcRGYw3TcbrcmTJigpKQkLV++XKWlpZo+fbquueYa+Xw+eb3eoBQVJnUCAKLF4XAEzWlas2aN\ntm7dqqqqKpWXl2v69OmqqalRUVFR0L2LQSWEQmAO0/DPbM/KytKdd96pFStWaOXKlVq9erWuueYa\nWSyWoNGFlqt5AgBwrrUcHMrLy1N2drYyMzNls9n0wAMPaPny5Tp9+rQqKirafB0gkWMOE/E/1nvx\nxRe1YsUKSdKqVasCiwm1HF1g9jsAwCya54z7g26Px6Ply5dLkq6++uqQxwLNEZjDNPyBd1FRkVav\nXq0777xTSUlJKikpUXZ2dqtKKzwGBACYRajBIofDoZKSEknSxx9/HAjE3W53yFWqAVJZYAot01JS\nUlK0cuVKTZ48WTNmzAiZqsJjQACAWYQaLLJarYHAvLKyUhkZGcrOzlZjY6PuuusuVVRUBC2EBzBi\nDlNoaGhQTk5OoBqLw+FQWVmZli1bRhlEAIDphVo/wz/oNGbMGCUlJcnn86myslITJ04MFDew2Wyk\ntSCAwBxR07xG+datWwO55JJYvRMAEPP86S0Wi0V1dXWyWCyqrKyUJM2cOVNFRUUha5wjcZHKgqhp\naGhQfn6+iouLNXv2bJWXlys7Ozuwn1QVAEAsa57eYrVa5fV6VVJSoqqqKl1//fXyeDxyOp2aPHly\n0OrVSFwE5oi6f/7zn1q5cqVyc3MpewgAiBvNB5j8aS0zZsxQZmZm4GlwWVmZnE6n7rnnHgajQCoL\nzo1QpaHsdrtuvPFGrVixQqtXr45i6wAA6F7+tBaPxxPIRSdtEy0RmOOcaFlGyuv1qqamRps2bdJP\nfvITLV++nPw6AEDcaqvEb6hJo0hcBOaIuJaj416vV42NjZo1a1Ygh87tdsvpdOruu+/WY489JpfL\nxWgBACButVW1peX9kgotiY3AHBHXcnTc7XZr0qRJqqqq0vr161VfXy+bzabi4mK98cYbuuuuu2Sx\nWBgtAAAkDK/Xq+rqauXk5ATuly1LBzc/loA9MRCYI+JaPq5zOBxas2aN1qxZI0mBHLuMjAxJUklJ\nCaPlAICE4na7VVFRobKysrPeA0OtKor4RFUWRFzzWegnTpxQTU2NioqK1KtXL3m93qCg3T/hhdFy\nAEAiaVlKUZKys7NDTgRtOeDVcrVsxA9GzBExoR611dTUqLy8XDU1NZKCc+yY8AIASFSh7oFt3Rdb\nbmcEPX4xYo6I8V8oXC6XLBaLHA6HioqKJCnwXwAAEFpHR8IdDodcLpd8Pp+8Xi8DXHGEEXNEjP9R\nm3Qmj7yhoUEej0czZ85Ur169otw6AADMraMj4VarVRaLRQUFBYyaxxkCc0RE80/52dnZQQE6Fw0A\nAM6urVrnXT0WsYPAHBERqsST3W7nogEAQAeFM/eKeVrxicAcndJyoqfP55NhGPL5fCGXHQYAAJ1D\nHfPEQWCOTmmZB2exWJSUlBSY9FlbWyubzcaFBACATvJ6vVq7dq0qKytJDU0QVGVBp7TMbWtee9X/\neK2+vl65ubmqra0N1DUHAAAd43a7lZ+fL8MwVF5eTmpoAiAwR9hClXOyWq1yOBxB25mYAgBA5/lX\nzpbODIBZrVZ5vd7AfC7/NolFh+IFqSwImz+Npbq6OihNxe12KycnJ7CdiSkAAHSe1WpVXl6e8vLy\nghYXys/PV35+flBqC4sOxQcCc4TN4XCotLRUTqcz6ALgcDhUVlamiooKLgwAAHQD/yj6mjVrAk+k\nvV6vfD6fXC4XT6ljHKksCJvVatXMmTM1duzYoAtAW9sBAEBk+EfRm3O73SooKFBtbS1PqWMcgTk6\nxZ+m0tHtAACgezCnK36QygIAABCjmPQZXwjMAQAAYlRbkz5ZlCg2EZgDAADEqFBpLF6vV9XV1crJ\nyaEYQ4whxxxn1VbNVAAAEF2h5na53W5VVFSorKyMvPMYw4g52uX/1D1hwoRWNVMBAIA5NE9d8Y+i\nz5w5k8G0GENgjjb5g/KKigqVlJRo1apV8vl85KsBAGAyzXP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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf040c3c8>" ] }, "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='Sensor', color='k', marker='.', s=2)\n", "xs, ys = sum(xs)/N, sum(ys)/N\n", "plt.scatter(xs, ys, c='r', marker='*', s=200, label='Mean')\n", "plt.scatter(0, 50, c='k', marker='o', s=300, label='Intuition')\n", "plt.axis('equal')\n", "plt.legend();" ] }, { "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. Even if it doesn't diverge the solution will not be optimal. Linear approximations applied to nonlinear problems yields inaccurate results." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Effect of Nonlinear Functions on Gaussians\n", "\n", "Gaussians are not closed under an arbitrary nonlinear function. Recall the equations of the Kalman filter - at each evolution we pass the Gaussian representing the state through the process function to get the Gaussian 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 let's use sampling. I will generate 500,000 points with a normal distribution, pass them through $f(x)$, and plot the results. I 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": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf07a27f0>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "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": 4, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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QWm3CF7sz8OWe8xCvO6bbeLXz0oncUpz46QT+tSUN3Vt5oajSCKtYW7o2rmtL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\nYyIYKRERUFpaioKCgk7bDhw4gNzc3DBHFB9EUcL1Zjeqmt1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RERERRRkm6UREREREUYZJ\nOhERERFRlGGSTkREREQUZZikExERERFFGSbpRERERERRhkk6EREREVGU+f+ekpUGXMKzUAAAAABJ\nRU5ErkJggg==\n", "text/plain": [ "<matplotlib.figure.Figure at 0x1bcef378b00>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from kf_book.book_plots import set_figsize, figsize\n", "from kf_book.nonlinear_plots import plot_nonlinear_func\n", "\n", "def g1(x):\n", " return 2*x+1\n", "\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 function $f(x)=2x+1$ which is displayed in the chart on the bottom left. 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 right. 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 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": {}, "outputs": [ { "data": { "image/png": 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pTG2ubl9frfYiVUygr4WLooJ5/fqLyS20k5ZrY9p32/liw8ES69nsxderWLMn\njTV70pgaHkDLusGk59pwAY3DAxjeqSHdm+nXORERqTxK8s9Ceq6NLzYcYE9KLlHH+8cfzMjn94NZ\n5BXaOZpdSGZ+EQCHswpKbV8rwEqj8ACCfC2M6daEAXF19WUvUo0E+loI9LXw8qiOPDKoNflFDpJT\nc3nn5z0cziqgoMjB/vR8APam5ZUYZ7M+OZ3PfztArQArXZtG0KpeMC2jgmkSEQhAWICVeqF+6u4j\nIiIVqkYn+Zl5RRzOKsDhdOF0uSgocnAsu5AdR3JITsvFbBgkp+WxYW9GiavLnorZZOBwugj2tdCn\nVSRhAVaiIwK5tnMjgnxr9FMt4jXqhhaf6DetHUh8y0ig+IrUK/9I5Y1lu/l5Z+npbwHS84pYtPkw\nizYfLrUsyNdCqL+V8EAfmtYOJLZ+CG3qh+JnNREW4EPj8ABN5ykiIuekRmSej8zbiNlkJizASr7N\nQU6hnb1peaxLSqOMa0ydE3+rGbvTSf/Yujw9pA0uir+w9YUsUnMYhkGP5rXp0bw2R7MKcLogIsgH\nh9PFD1uPMj/xACt2p5JdUHY3vpxCOznHxwBsOpDJ/MSDpdbxMZvwsRz/M5swmwwK7Q6a1Q7k7evi\nLvQhiohINVMjkvzFm4+Qnn/qPvJn0jg8gP6xUVweG0V6rg2TySAi0IeWdYMJ9rPidLrU/UZEANxT\n3gJYzTC4XT0Gt6uHw+kiKTWXnUey2X44h4MZ+ZhMxWN69qTkknO83/+pGh5sDmfxL4p/uZSG4yx+\nZRQRkZrHcLlONedL9eR0OsnOzi5RFv/yqjKT/PrBFlrV9sFqMjAMsJoMQv1M1A2y0DjUimFAkI+J\ncH9zZYV/WkVFRe7bVqt3zMLjbcfkbccDOqbKVORwcSDbzo5UGwezinC4ID3fwdFcBzaHiyKniyKH\nC7sTHC4XVpOBE4MlE7qV2E9wcDAmk35NFBGpyWpEkr9y0w4y8+3k2pz4WUz4Ww2CrAZhfiYNdhOR\nas3mNIhrGVOiTEm+iIh4XXedss5ZGtUKoEFo9f9J227/89cIi8U7XjpvOyZvOx7QMVV1/mUk817W\ndiMiIuVQvb/dylDWl1vDhg09EImIiGcoyRcREf2eKyIiIiLiZZTki4iIiIh4GSX5IiIiIiJexitn\n13E6Sw6yNQxDs+iIiFdyuVyl+uCbTCbNriMiUsN5XZIvIiIiIlLTqalHRERERMTL1Jgkf8OGDQwe\nPJjGjRvj7+9PeHg43bt3Z/bs2Z4OrVyWLFnC+PHjadWqFYGBgTRo0IBrrrmG9evXezq0csvOzuZf\n//oX/fv3p06dOhiGQUJCgqeOh/sDAAAgAElEQVTDOis5OTnce++91K9fHz8/Pzp06MBHH33k6bDO\nS3V+PcrijZ8Zb6vXRESk4tSYJD8jI4NGjRrx7LPPsnDhQt577z2io6MZM2YMTz/9tKfDO2dvvPEG\nSUlJTJgwgYULF/LKK69w9OhRunXrxpIlSzwdXrmkpqby1ltvUVhYyJAhQzwdzjkZNmwYs2bNYtKk\nSXzzzTd07tyZ6667jg8//NDToZVbdX49yuKNnxlvq9dERKTi1Pg++d26dePgwYPs3bvX06Gck6NH\njxIZGVmiLCcnh5iYGNq0acP333/vocjK78Rb0TAMUlJSqFOnDpMmTaryrccLFy5k8ODBfPjhh1x3\n3XXu8v79+7N582b27t2L2Wz2YITlU11fj1Pxxs/MqVTXek1ERCpOjWnJP5XatWtXy8va/zVZAQgK\nCiI2NpZ9+/Z5IKLzV11nQfr8888JCgpixIgRJcrHjRvHwYMHWb16tYciOz/V9fU4FW/8zJxKda3X\nRESk4tS4JN/pdGK32zl27BjTp0/n22+/5eGHH/Z0WBUiMzOTX3/9lbi4OE+HUqP8/vvvtG7dulRS\n1a5dO/dyqZq85TPjzfWaiIiUT41r6rnjjjv473//C4CPjw//+c9/uO222zwcVcW48847yc3N5fHH\nH/d0KDVKamoqzZo1K1UeHh7uXi5Vk7d8Zry5XhMRkfKpli35y5Ytc3clONPfhg0bSmz72GOPsXbt\nWr7++mvGjx/PXXfdxdSpUz10JMXO53hOeOKJJ/jggw946aWX6NSpUyUfQWkVcUzVyem6tXhTlxdv\nUtU+M+ejKtZrIiLiWdWyJb9ly5a8/fbbZ7Vu48aNS90/UXbFFVcA8Oijj3LjjTdSp06dig30LJ3P\n8QBMnjyZp59+mmeeeYa77rqrosMrl/M9puokIiKizNb6tLQ04M8Wfak6quJn5nxUxXpNREQ8q1om\n+fXq1ePmm2+ukH116dKFN998kz/++MNjX4bnczyTJ08mISGBhIQEHnvssQqOrPwq8jWq6tq2bcuc\nOXOw2+0l+uVv2rQJgDZt2ngqNClDVf3MVKSqUK+JiIhnVcvuOhVp6dKlmEymMvtUV3VTpkwhISGB\niRMnMmnSJE+HU2MNHTqUnJwc5s6dW6J81qxZ1K9fn65du3ooMvmrmvKZqc71moiIVIxq2ZJfHrfe\neishISF06dKFqKgoUlJS+PTTT/n444956KGHql1r17Rp03jyyScZOHAggwcPZtWqVSWWd+vWzUOR\nnZ9vvvmG3NxcsrOzAdiyZQufffYZUNwNISAgwJPhlWnQoEH069eP22+/naysLGJiYpgzZw6LFi1i\n9uzZ1XKO/BOq4+txKt74mfG2ek1ERCpOjbkY1owZM5gxYwZbt24lIyODoKAg2rdvz80338zo0aM9\nHd45i4+P58cffzzl8ur6skZHR5OcnFzmsj179hAdHV25AZ2lnJwcHn/8cT755BPS0tJo1aoVjz76\nKKNGjfJ0aOelur4eZfHGz4y31WsiIlJxakySLyIiIiJSU9T4PvkiIiIiIt5GSb6IiIiIiJdRki8i\nIiIi4mWU5IuIiIiIeBkl+SIiIiIiXkZJvoiIiIiIl1GSLyIiIiLiZZTki4iIiIh4GSX5Iqfx+uuv\nExcXR0BAAIZh8PLLL7uXPfnkk/j5+bF///5y73/evHkYhsEPP/xQEeGKiIiIAEryRU7pk08+4a67\n7sLX15cJEyYwadIkunXrBsC+ffuYOnUqt99+Ow0bNiz3YwwbNoyLL76Y+++/H6fTWVGhi4jUGOvW\nrWPcuHE0a9YMf39/QkJCaNu2LQ899BAHDhyokMcYO3YshmGQlJRUIfsrj2XLlmEYBgkJCR6LQaoX\ni6cDEKmq5s+fD8CCBQuoX79+iWVTpkzBZrPxr3/967wf55FHHuHaa6/lo48+4vrrrz/v/YmI1AQu\nl4tHHnmE559/HovFQr9+/RgxYgQ2m40VK1YwdepUpk+fzqxZsxg+fLinwxWpdEryRU7h4MGDAKUS\n/MzMTD744AMGDBhAvXr1zvtxrr76amrVqsX06dOV5IuInKWnnnqK559/nujoaBYsWEBcXFyJ5XPn\nzmX06NGMGjWK7777jr59+3ooUhHPUHcdkb9ISEjAMAyWLl0KgGEY7j+AOXPmkJeXx8iRI0ttO2TI\nEAzD4NVXXy217IknnsAwDG677bYS5b6+vgwZMoTly5ezbdu2C3BEIiLeZc+ePTz99NNYrVbmz59f\nKsEH+Pvf/85LL72Ew+Hg9ttvd3eJPFHHL1u2rNQ2SUlJGIbB2LFj3WWGYTBr1iwAmjZt6v4+iI6O\ndq8THx+PYRgUFhYyceJEmjZtiq+vL82bN2fy5MnYbLYzPs7JTuzvhLFjx9KnTx8AJk+eXOJ7qazj\nEAG15IuUEh8fD8DMmTNJTk5m0qRJJZYvXrwYgB49epTa9t1336Vjx4489NBDXHrppXTs2BGAH374\ngWeffZY2bdqUGLx7Qo8ePZgxYwaLFy+mVatWFXxEIiLeZcaMGdjtdkaMGEHbtm1Pud7NN9/MU089\nxY4dO/jxxx/difK5mDRpEl988QWJiYlMmDCBsLAwAPf/k1177bWsXbuW4cOHY7Va+fLLL0lISGDd\nunXMnz+/ROJ+LoYMGQLArFmz6N27t/t7CihxsiFyMiX5In8RHx9PfHw8y5YtIzk5udQgp+XLlxMa\nGkpMTEypbcPDw5kzZw69e/dm5MiR/Prrr+Tl5TF69Gj8/Pz4+OOP8ff3L7Vd586dAfjpp5+4++67\nL8hxiYh4i19++QWAfv36nXY9i8VCfHw8c+bMYfny5eVK8hMSEkhKSiIxMZF77733tEn11q1b2bx5\nM7Vq1QLgmWeeoU+fPixYsIDZs2czZsyYc358KE7yw8LCmDVrFvHx8Rp8K2dF3XVEzoHNZuPIkSNE\nRkaecp0ePXowZcoUdu7cyW233cbo0aM5fPgwr776KrGxsWVuU7duXaB41h4RETm9w4cPA9CoUaMz\nrntinRPjrC6kJ554wp3gA/j5+fHcc88Bxb/0ilQmteSLnIPU1FSAEpV4WR5++GGWLVvGhx9+CMB1\n113H+PHjT7l+eHg4ACkpKRUUqYiI93K5XABn1f3lxDrl7SpzLnr37l2qrFevXlgsFn777bcL/vgi\nJ1NLvsg5ONHVpqCg4LTrGYbB0KFD3ffvvffe066fn59fYv8iInJqJ2Y2O5tfP09csLAiZkM7k6io\nqFJlZrOZiIgIsrKyLvjji5xMSb7IOQgLC8PHx8fdon8qO3fu5MEHH6RWrVqYTCZuueWW054YnNjf\n6boBiYhIsUsvvRSA77///rTrORwO90xpPXv2BMBkKk597HZ7qfUzMjLOK64jR46UGUNqaiohISHu\nstPFUBFxiICSfJFz1rZtWw4dOnTKVpnCwkJGjhxJbm4uH3/8MY8++igbN27kvvvuO+U+T0yd2aFD\nhwsSs4iINxk7dixms5l58+a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OncvcuXNp0KAB/fv3p3///sTExHg63DP660lMdXRyteIN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\n9rds3LiR7OxsDh48yOLFixk6dCgKhYKkpCQWL15MTEwMS5YscXeTPV6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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf3316a90>" ] }, "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", "plot_nonlinear_func(data, g2, gaussian)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This result may be somewhat surprising to you. The function looks \"fairly\" linear, but the probability distribution of the output is completely different from a Gaussian. Recall the equations for multiplying two univariate Gaussians:\n", "\n", "$$\\begin{aligned}\n", "\\mu &=\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2} \\\\\n", "\\sigma &= \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n", "\\end{aligned}$$\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": {}, "outputs": [ { "data": { "image/png": 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FQ383pcMiIiKi2zBJJ8X9EZWEb7fF4GxiFgBr1xWjpeS4+D6uGjzesg5GdghG2/reVR1m\njdGnZQBOXM0AAMz78wIWjGyncERERER0OybppKg1R+LxwdrTdmVFCXqzAHe83zcUBrMFoXU80cjf\nDaJYtgfw0C0vdGyA/+yJRWquEZtOJ+LlbhkI50kPERGRQ2GSTpXGbJGgVtkPnZmZZ8KBy6k4cTUd\nl1Nyse3cDdu8tvW8AEHAqfgM1PV2wY+jH0Y9n/I/cIfseeg1mPBYU0xdHwUA+HLLeaz8e0eFoyIi\nIqLimKRTpdgVfRPjVhyDt6sWEx5risdaBOC30wn4YvN5u5s/i4zpEoKpT7eEIAi4npEPH1cNXLX8\nelaWEY8E48e9sbiSmof9l1KRkJGPIG/eeEtEROQomAVRhcszmjFp7WnkGi3INebjg19O33FZrVrE\ny10b4v2+zSEI1q4sdZksVjqNSsSgiHr4els0AODPqCSM7tJQ4aiIiIioCJN0qjCHY9Ow72IKrqXn\nIzGz4I7LDW1fD33D6iC0jgcCvfQlusRQ1XiiVR1bkr6FSToREZFDYZJOZXbkShpeX3EcIX6ueLlb\nI7yx8rjdU0A1KgGzh7TB4dg0JGYWQKsW8VKnEHRt6q9g1FSkWYA7QvxccSU1D4dj05CWa4Svm1bp\nsIiIiAhM0ukByLKMhMwCFJgsCPDUY+Lqk0jJMSAlx4CjccdKLD+6cwgGt6tnG8+cHIsgCOjbqg4W\n7boMSQa2nbuBZx+qr3RYREREBCbpdBe5RgmbTidg38VUnEvMwsXkHOQYzACsY5anl/KwoXbB3mhT\nzxvuOjXe7N2kqkOmB9Q3zJqkA8B/9sRiQHgQdGqVwlERERERk3QqVVKOGdN2pSA1P6HU+UUJul4j\nopaHDvFp+Qjw1OH7F9qjtoe+KkOlcgiv542WgZ44m5iFCzeyMX/7RbzXt7nSYRERETk9JulOLrvA\nhDPXM3E1NQ++blp0auwHWZbx/bEspObbD5VYz8cFzQI8kJRZYHs66Pt9QzGkXV38EZWEbk1rMUGv\nZkRRwJxhbTBgwT6YJRnf7bqEfq3rICzIS+nQiIiInBqTdCeVb7TgP3sv47vIS8gtNm65WhRQz1ON\nKxnWlvK63i74YnBrtG/gAzed9esiSTL2X0qFWZLQo3ltAMBzDwdX/YegChEW5IU3ezXF19uiYZFk\nrDh4FV8Mbq10WERERE6NY985oesZ+ei/YC/m/hltl6ADgFmSbQk6AMwc3BqPNqtlS9ABa+tr16b+\ntgSdqr+XuzWEi8baF/33M4kwmiWFIyIiInJubEl3EjeyCvBd5CVcS8/D6WuZSM42AABUooCB4XXR\nuq4nYlNysSv6Jq6k5gEAHm/shu7NaikZNlURN50afVoGYMOpBGTmm7Ar+ib6tAxQOiwiIiKnxSS9\nBpNlGTkGMy7dzMX4n47jeka+3fyG/m5Y9EJ7NAvwsCvfsvcYUnIMCKvjWpXhksIGRgRhwynrjcLr\nT15nkk5ERKQgJuk10LazNzBtQxSSsgpgKfZwoeIeaeiL755vBz93XYl5gR5q+OtliIJQ2aGSA+nW\ntJZtaM2tZ2/gZrYBtTxKfj+IiIio8jFJr2HyjGZM+vU0UnKMJeaFBXni2+ERcNepUceLo7CQPY1K\nxDNtg/DfA3EwmCW8s+Yklo15BKLIkzUiIqKqxiS9hlm2P86WoAd56dGolju8XDUIDfDAmK4N4a7j\nPznd2Zu9muL3M0lIyTFgT0wKvt99CeN68KFUREREVY0ZWzWWlmtEbEouTBYJ4fW9YTBL+H7XJQCA\nKAD//dsjaFLb4x5bIbqllocO3w4Px6j/HIIsA99si8HA8LoI8nZROjQiIiKnwiS9mvrXzov4ems0\nzIV9zuv7ukCrEpGZbx0+cWBEXSboVCZdmvhjbJeG+M/eWBjNEr7dFoPZQ9soHRYREZFT4Tjp1dCK\ng3GY88cFW4IOAPFp+bh0MxcA4KFT4+3HmikVHtUAb/ZqAg+99Rz+f8ficTE5W+GIiIiInAuT9Grm\nz6gkTF3/l216UERddGzka5sOC/LE/43vgvq+HD6Rys7bVYvXujcGAEgy8MEvp/mAIyIioirE7i7V\nyLG4dLy56gSKGtBf7d4IH/VrAVmWsTsmBSnZBjzdNhA6tUrZQKlGGNulIVYeuorrGfk4fjUD0zdG\nYeag1kqHRURE5BTYkl4NSJKM1YevYvSPh2EobM0cFFEXH/YNBQAIgoDuzWphSPt6TNCpwrhoVfhu\nVDto1dbDxMpDV/H7mUSFoyIiInIOTNIdXFaBCWOXHcGkX88g22AGAHRt4o/ZQ9pw/GqqdG3qedu1\nnv/jt3MwWkp/QBYRERFVHCbpDiwhIx9DFu5H5IWbtrLB7eri+xfa21o3iSrbkHZ10a2pPwDgekY+\nNpznTaRERESVjZmeg5IkGW+uOoGY5BwAgI+rBsvGPoKvng3nA4moSgmCgMlPtUTRhZs1Udk4k1zy\nibZERERUcZikO6i1x6/hWFw6AKCejwv+b1wXdG9WS+GoyFk1r+OB5zs0AAAYLDL+sTcdB+LzFY6K\niIio5mKS7oAy80yYtfm8bXrW4DYI8XdTMCIi4JOnWqB3aG0AgFkCvjqQinOJWQpHRUREVDMxSXdA\n83fEIDXX2p3gqdaB6FrYH5hISXqNCoteaI/uDaxj8BstwLifjiO7wKRwZERERDUPk3QHE5+Wh/8e\niAMA6NQiPnmqhcIREd2iVol4o4MPGnpb74uITcnF0O8OIPoGbyYlIiKqSEzSHcxXW6NhtFjHQh/b\ntSGCvF0UjojInlYl4N2OXnDTWO8kvXAjGwMW7MPZBHZ9ISIiqihM0h3IvospWHfyOgDraC6v92is\ncEREpavjrsasPrXRPMADAJBvsuDTTVGQZY6hTkREVBGYpDuI5KwCTFh9AkU5zlu9m8JTr1E2KKK7\nCPbSYP0bXRDiZ+2jfvByGrafS1Y4KiIiopqBSbqDeO+X00jJsd4s2r1ZLbzUKUTZgIjug16jwodP\nhNqmZ/5+Dlm8kZSIiKjcmKQ7gL+uZ2J3tPWponU89fjq2bYQi54cQ+TgnmhVB+0b+AAALqfk4oV/\nH0JmHhN1IiKi8mCS7gCWF47mAgBv9m4CP3edgtEQPRhBEDBrcGv4umkBAKeuZWLQd/s44gsREVE5\nMElXWEae0XazqIdejYHhdRWOiOjBNQ3wwOpXOsK/8ATz8s1cDFiwD5vPJCocGRERUfXEJF1hq4/E\nw2C2Drk4rH19uOnUCkdEVDbNAjyw9vVOaBHoCcA64su4lcexZF8sJImjvhARET0IJukKSs4qwL92\nXrRNj+oYrGA0ROXXwM8N/zeuMwZHWK8IyTIwY+NZPPnPPfgzKknh6IiIiKoPJukKmrHxLLILzACA\nIe3qoVEtd4UjIio/vUaFec+2xfiet8b5P5+UjVeWH8MHv5xCcnaBgtERERFVD0zSFfJnVBJ+K+yv\n6+OqwSdPtVA4IqKKIwgC3u8biiVjHkZ4fW9b+Zqj1/DIP7ajz1e78PORqzAXPl2XiIiI7DFJV8DN\nbAM++vWMbXryUy1tI2MQ1SQ9m9fG/43rjLnD2sJVq7KVxyTn4MO1Z9D3m92ISshUMEIiIiLHxCS9\nismyjElrTyM11/rgosdaBGBwO47oQjWXIAgY2r4e/pj4KMb1aGzXsn7pZi4GL9yPb7ZF49DlVFh4\ngykREREAgEOJVLFVh+Ox/bz10en+7lrMGtIagsAHF1HNV9/XFR8UPp30WFwapm84izPXM2EwS/hm\nWwy+QQza1vfG/OERCPZzVThaIiIiZbElvQrFpuTis01nbdOzh7SxjStN5EzaN/DFL693wsgO9iMa\nnYrPwJP/3IMZG6NwMj6DfdaJiMhpsSW9imQVmPDGyuPIN1kAACMeCUbvFgEKR0WkHJ1ahZmDWuPv\n3RrhSGwaFkZexJXUPOQYzFiy7wqW7LsCF40Kz7QNxCdPtoSXq0bpkImIiKoMk/QqkGswY8ySI4hK\nyAIAhPi5YjJHcyECADT0d0NDfzc82SYQ//jtHNYevwZj4QO+8k0WrDl6DXtiUjCmSwhC/NwQl5oH\nHzct+oYFwEPPxJ2IiGomJumV7FxiFt5adQIxyTkAAF83LRa/+BCfLEp0G3edGl8Mbo1JT4TitzOJ\nOHA5FZHnk5FtMCMxswAzfz9vt/yUddZW9mcfqo/QQE+4aVW8v4OIiGoMZoqVRJZlLD8Yh89/O2dr\nFfTUq/HfsY+gaYCHwtEROS4vVw1GdgjGyA7BSMjIx4drT2NPTEqJ5Ypa2dccvQbA+ryB/m2D0LdV\nHQR5uaCejwvUKt52Q0RE1ROT9EpwLT0PU9dHYUfhKC4A0CLQE/NHRKBJbT5VlOh+BXm74L9jH8Gl\nmznYfykVKdkG1Pd1xalrGVh/IgHZBrNt2fQ8E5YdiMOyA3EAAJ1aRMsgT7Su64XGtdyhVYtoUtsd\nDzXwYYs7ERE5PCbpFSgt14gl+2KxeM9lFJhujUoxpksIPnwiFHqN6i5rE1FpBEFAk9oeaFL71hWo\nYQ/Vx8dPtsBvpxOx92IKkrMMOBGfbvf/zmCWcOJqBk5czbDbXtt6XmhUyx15RjPa1PNGq7peMJgs\nqO/rihaBnlX2uYiIiO6GSXo5ZBeYcPByGvbG3MSpa5k4m5hl69oCAP7uOswe0pqjuBBVAletGsMe\nqo9hD9UHAGTmm/D7mURcTM5BYmY+ziZk4UpqXon1Tl3LxKlr1qec/hF1w25ey0BP1PVxgcEswU2r\ngodeDXedBkHeejSv44HmdTxQy12HPKMFkizzxlUiIqo0TNLvgyzLyDaYkZZjxI2sAmtifvEmjl/N\nKPUJiWpRwKiODfDO483gyR9xoirh5aLBiEfsx13PzDchKiETiRkFyDGYserwVZxPyr7jNs4mZuFs\nYtZd96NTizAUnoz7uWmhggVGswxXrYh6B/LQuYkfHg7xhQDAIsuQZesN496uGmTlm2GySHDRqqBX\nq6DXiNCp2W+eiIhKcookXZZlmCwyjBYJJrMEo0WCwSQhJdeAlGwDLsfnIyXXiIvpFgjHD0OjEpFj\nMCMt14jUXCPSc40w38fjykP8XNErNAB/69YQdb1dquCTEdHdeLlo0Lmxv236xU4NcOlmjq1P+oFL\nqUjMzIdGJWLn+WRbC/vdGIpdLUvNNdr+zjBISMhOw+EraQ8Uo4+LGpETOz7QOkREVPNViyT9xf8c\nQmaBBRZJhiQXvQCpcNoiy5Ak2M0zWyQYzZItOb9/JS+P30mjWm7o1sQf3ZrWwiONfNlqTuTgivq3\nF2lc69aN3BMfa4bk7AJAtj5oKddoRnaBGVkFJlxJycWFpGxcuJGNhIx825OCr6TmosBggkYF5Jtk\n5JrufTJPRER0PwRZlh3qV0WSJGRn21+O7vHNQaTnm++wRuVRi4CXTgVPnQgvvQhPnQgPnQoh3hqE\n19Ghtlu1OMd5YCaTyfa3RsMTD9aHPdaHveL1kWkScOR6AZJzLRAFQBQAGUBmgYRsowR3rQiNCBgt\nMowWGQaLDK1KhQUjI+y26eHhAVFkNxgiImdWLbJMAYBKAITCHz0RAsRi09Z3ASJulakEAWqVtX+4\nRhSgFmF7V4sCNCrAUyvCWy9CJQrQqQQ08dWgtpsKFkmGXi3ARS3cZag22e7HuaZyhs/4IFgf9lgf\n9rw0Mh4L0T3QOioVR30iIqKSHC5JL61hf/nQepCkB+my8mDM5lut9Gq1w1VJlWN92GN92GN92Ctv\nfZTWYu5gFziJiEgBDvcLW9qPU7169RSIhIhIGUzSiYiInR6JiIiIiBwMk3QiIiIiIgfDJJ2IiIiI\nyME45BCMt98kKgh3G2WFiKj6kmW5RB90URQ5BCMRkZNzuCSdiIiIiMjZsamGiIiIiMjBMEm/h3//\n+98QBAHu7u73XrgG2rFjB8aOHYvQ0FC4ubmhbt26GDBgAI4dO6Z0aJUuJycHEydORFBQEPR6PcLD\nw7F69Wqlw1KEM38P7oezHyeIiKjisbvLXVy/fh1hYWFwc3NDZmYmcnJylA6pyg0bNgypqakYNmwY\nWrZsiZs3b2LevHk4evQo/vjjD/Tq1UvpECvN448/jiNHjmDWrFlo1qwZVq5ciX//+9/46aefMHLk\nSKXDq1LO/D24Fx4niIioMjBJv4tnnnkGgiDA19cXv/zyi1P++CYnJ6N27dp2ZTk5OWjSpAlatWqF\nbdu2KRRZ5fr999/x1FNPYeUQ8BscAAAgAElEQVTKlRgxYoSt/PHHH0dUVBSuXr3qVI9zd9bvwf3g\ncYKIiCoDu7vcwYoVK7Br1y4sXLhQ6VAUdXtiBgDu7u5o2bIl4uPjFYioavzf//0f3N3dMWzYMLvy\nMWPGICEhAYcOHVIoMmU46/fgXnicICKiysIkvRTJycmYOHEiZs2ahXr16ikdjsPJzMzE8ePHERYW\npnQoleavv/5CixYtoFar7crbtGljm+/snOF7cDc8ThARUWVikl6KcePGoXnz5nj99deVDsUhjR8/\nHrm5ufjkk0+UDqXSpKamwtfXt0R5UVlqampVh+RwnOF7cDc8ThARUWWq0Ul6ZGSk7UFI93qdPHkS\nALB27Vps3LgRixcvrnEPUCpLfdxuypQp+Omnn/D111+jffv2VfwJqtbd/v1r2nfjQTnT96A0Nfk4\nQUREjkF970Wqr+bNm2Px4sX3tWxwcDBycnIwfvx4vPnmmwgKCkJGRgYAwGg0AgAyMjKg0Wjg5uZW\naTFXpgetj9vNmDEDn3/+Of7xj3/gjTfeqOjwHIqfn1+preVpaWkAUGoru7Nwpu9BaWr6cYKIiByE\nTDaxsbEygLu+BgwYoHSYipg+fboMQJ4+fbrSoVSJv//977K7u7tsMpnsyletWiUDkPft26dQZMpy\ntu9BaXicICKiqsAhGIspKCjAwYMHS5TPmjULu3btwubNm+Hv749WrVopEJ1yPvvsM0ydOhWTJ0/G\nZ599pnQ4VWLz5s148sknsXr1ajz33HO28n79+uH06dNONwQj4Jzfg9LwOEFERFWBSfp9GD16tNOO\nfzxv3jy89957eOKJJzBt2rQS8zt27KhAVFXj8ccfx9GjRzF79mw0adIEq1atwuLFi7FixQo8//zz\nSodXpZz5e3C/nPk4QUREFa9G90mn8tu4cSMAYMuWLdiyZUuJ+TX5HO/XX3/FJ598gqlTpyItLQ2h\noaFYtWoVhg8frnRoVc6ZvwdERERKYEs6EREREZGDqdFDMBIRERERVUdM0omIiIiIHAyTdCIiIiIi\nB8MknYiIiIjIwTBJJyIiIiJyMEzSiYiIiIgcDJN0IiIiIiIHwySdiIiIiMjBMEmnGu1f//oXwsLC\n4OrqCkEQ8M0339jmTZ06FXq9HteuXSvz9n/99VcIgoDt27dXRLhEREREAJikUw22Zs0avPHGG9Dp\ndJgwYQKmTZuGjh07AgDi4+Mxd+5cvP7666hXr16Z9zF48GC0a9cO77zzDiRJqqjQiYicxtGjRzFm\nzBg0atQILi4u8PT0ROvWrfH+++/j+vXrFbKP0aNHQxAEXLlypUK2VxaRkZEQBAHTp09XLAaqXtRK\nB0BUWTZs2AAA2LRpE4KCguzmffbZZzAajfjggw/KvZ9Jkybh2WefxerVqzFy5Mhyb4+IyBnIsoxJ\nkybhyy+/hFqtRp8+fTBs2DAYjUbs378fc+fOxcKFC7Fs2TIMHTpU6XCJqhyTdKqxEhISAKBEgp6Z\nmYmffvoJffv2RWBgYLn3079/f/j4+GDhwoVM0omI7tOnn36KL7/8EiEhIdi0aRPCwsLs5q9duxaj\nRo3C8OHD8eeff6JXr14KRUqkDHZ3oRpn+vTpEAQBO3fuBAAIgmB7AcCqVauQl5eH5557rsS6AwcO\nhCAImD9/fol5U6ZMgSAIePXVV+3KdTodBg4ciH379uH8+fOV8ImIiGqW2NhYfP7559BoNNiwYUOJ\nBB0AhgwZgq+//hoWiwWvv/66rUth0TE+MjKyxDpXrlyBIAgYPXq0rUwQBCxbtgwA0LBhQ9vvQUhI\niG2ZHj16QBAEGAwGTJ48GQ0bNoROp0Pjxo0xY8YMGI3Ge+6nuKLtFRk9ejR69uwJAJgxY4bd71Jp\nn4MIYEs61UA9evQAACxduhRxcXGYNm2a3fytW7cCADp37lxi3R9//BERERF4//330bVrV0RERAAA\ntm/fjpkzZ6JVq1Z2N58W6dy5M5YsWYKtW7ciNDS0gj8REVHNsmTJEpjNZgwbNgytW7e+43Ivv/wy\nPv30U0RHR2PXrl22RPdBTJs2DevWrcOpU6cwYcIEeHt7A4Dtvbhnn30WR44cwdChQ6HRaLB+/XpM\nnz4dR48exYYNG+wS7wcxcOBAAMCyZcvQvXt32+8UALuTBSI7MlEN1b17d7m0r3hAQIDs5eV1x/X2\n7dsnq9VquWnTpnJ2drZ848YNuU6dOrKrq6scFRVV6jonT56UAchDhw6tsPiJiGqqnj17ygDkH374\n4Z7LjhgxQgYgf/bZZ7Isy/K0adNkAPLOnTtLLBsbGysDkF966SW78pdeekkGIMfGxpa6j6Lfi6ZN\nm8ppaWm28vz8fLljx44yAPm///3vPfdz+/aK27lzpwxAnjZt2j0/M5EsyzK7u5BTMRqNuHHjBmrX\nrn3HZTp37ozPPvsMMTExePXVVzFq1CgkJSVh/vz5aNmyZanr1KlTB4B11BgiIrq7pKQkAED9+vXv\nuWzRMkX3GVWmKVOmwMfHxzat1+vxxRdfALBeaSWqSuzuQk4lNTUVAOwOwqX58MMPERkZiZUrVwIA\nRowYgbFjx95xeV9fXwBASkpKBUVKRFRzybIMAPfVfaRombJ2NXkQ3bt3L1HWrVs3qNVqnDhxotL3\nT1QcW9LJqbi4uAAACgoK7rqcIAgYNGiQbXrixIl3XT4/P99u+0REdGdFI2vdz9XHogfOVcRoXPcS\nEBBQokylUsHPzw9ZWVmVvn+i4pikk1Px9vaGVqu1tajfSUxMDN577z34+PhAFEX8/e9/v2tiX7S9\nu3WjISIiq65duwIAtm3bdtflLBaLbaSuLl26AABE0Zq6mM3mEstnZGSUK64bN26UGkNqaio8PT1t\nZXeLoSLiIAKYpJMTat26NRITE+/YKmIwGPDcc88hNzcXP//8Mz766COcPn0ab7/99h23WTT0Ynh4\neKXETERUk4wePRoqlQq//vorzp49e8flfvzxRyQkJKB58+a2rihF3RVLa4U/evRoqdtRqVQArAn3\n3ezatatE2Z49e2A2m22jfd0rhqysLERHR5c5BqIiTNLJ6fTo0QOSJOHw4cOlzn/vvfdw4sQJfPjh\nh+jTpw9mzJiBLl264Pvvv8cvv/xS6joHDx4EgDIND0ZE5GwaNWqEjz/+GCaTCc8880ypifq6desw\nYcIEqFQqLFy40NZ63aFDBwC3hnEsEh8fj08//bTU/fn5+dmWuZvPPvsM6enptumCggJ89NFHAIAx\nY8bYyj08PNCiRQvs27fPLnaLxYJ33nnH1gWyLDEQFRHkors3iGqYHj16YNeuXbj9K37gwAF07twZ\n7733HubMmWM3b926dRg0aBA6deqE3bt3Q6223lsdHx+P8PBwWCwWnDx5ssS4tp06dcL58+dx/fp1\nuLq6VurnIiKqCSRJwvvvv4+vvvoKarUaffv2RVhYGEwmE/bv349Dhw7BxcUFy5Ytw7Bhw+zW7dmz\nJyIjIxEREYFevXrhxo0b2LhxI/r27Ys1a9bgpZdewtKlS23L//HHH3jiiSfQpEkTDBkyBO7u7vD2\n9sYbb7wB4NbvxYABA3D48GG7cdIvXbqEp556Chs3brS7eXXZsmUYPXo0vL29MWzYMOj1euzcuRMm\nkwl6vR6nTp2y+/2xWCxo0KABUlJS8MILLyA4OBiCIOCFF15AgwYNKreyqXpSdABIokp0p3HSZVmW\nIyIi5MDAQNlsNtvK4uLiZB8fH9nb21u+cuVKiXXWrVsnA5A7dOggG41GW/mFCxdkAPKECRMq/kMQ\nEdVwhw4dkl988UU5JCRE1uv1spubmxwWFia/++67cnx8fKnrZGRkyK+88opcq1YtWavVymFhYfKi\nRYvuOn75vHnz5NDQUFmr1coA5AYNGtjmFf1eFBQUyJ988okcEhIia7VauWHDhvL06dPlgoKCUuP4\n8ccf5ZYtW8parVYOCAiQX3nlFTklJeWOvz+HDx+We/XqJXt6esqCINxxvHciWZZltqSTU1q1ahVG\njhyJX3/91W4Ul7J49913sWDBApw7dw6NGjWqoAiJiKiq3OnKK5GSmKSTU5JlGZ06dUJ+fj5OnjxZ\n5vF3ExMT0bhxY4wbNw5z586t4CiJiKgqMEknR8QbR8kpCYKAH374AYMGDSrXU+yuXLmCDz/8EJMn\nT67A6IiIiMjZsSWdiIiInBpb0skRMUknIiIiInIwaqUDuJ0kSZAkya5MEIQy9xkmInJksiyXaL0T\nRdE2JnRNxuM9ETmLshzrHTJJz83NVToMIiLFuLm5OU2SzuM9ETmrex3ra/6vABERERFRNcMknYiI\niIjIwTBJJyIiIiJyMA7XJ720G4YefvhhpKWlVcj23d3dERAQgICAAAQGBiI4OBgqlQq1a9dGw4YN\n8eijjzr9TUtRUVEwm81Qq9UICwtTOpyqd/o08NhjwLZtQJs2rI/bsD7slbc+SuuX7SzHoNI+54P0\nxy+qe1GlwoITBuQazACAXi1qY2yXhhUaa03E/8vlw/orP2eqw7Ic66tFkv7ll18iOzsbBoMBRqMR\nBoOhxN/5+fkoKChAfn4+cnNzkZ2djezsbGRlZSE9PR1paWmwWCxITU1FXFzcHffv7e2Nli1bonXr\n1njooYfw0EMPISwsDBqNpjI/tkORJAkWi8VpRpgoQZKA1FTruyiyPm7D+rBXGfXhzEn6g9Rj8bp/\n9uFg/HN7DABg46lEjOoYAr1GVaHx1jT8v1w+rL/yc/Y6rHZJemn69+9f7n88WZaRlZWFpKQkJCYm\nIjExEdeuXUNsbCxOnz6N+Ph4XLt2DRkZGdi/fz/279+PRYsWAQD0ej06duyIRx99FN27d0enTp3g\n4uJSER+NiIgqQNv6Xra/JRnIN1qYpBNRtVYtkvSKIAgCvLy84OXlhebNm9vNO3XqFEwmE2RZhl6v\nR1RUFE6ePImjR4/i6NGjyMzMRGRkJCIjIwFYk/YePXqgX79+ePLJJ9GkSRMFPhERERXxcrG/2pme\nZ4SPm1ahaIiIys9pkvT7odVq0bp1a7Ru3RrDhw8HYL0Uc+HCBezZswe7d+9GZGQkrl+/ji1btmDL\nli2YMGECWrVqhSFDhmDIkCFo1aqV01yqJiJyFJdv2vf1nPn7Ofz7pYcVioaIqPycrwPQAxJFES1a\ntMArr7yCFStWID4+Hn/99Rfmzp2L3r17Q61W46+//sKMGTPQpk0btG7dGrNnz8a1a9eUDp2IyGlo\nVPY/ZzeyDApFQkRUMZikPyBBEBAWFoZ3330X27ZtQ3JyMpYtW4b+/ftDp9MhKioKkyZNQnBwMPr0\n6YM1a9bAaDQqHTYRUY3WpLY7Ajx1dmUFJotC0RARlR+T9HLy8fHBiy++iPXr1yMpKQmLFy9G9+7d\nIcsytm3bhueeew716tXDBx98gNjYWKXDJSKqNJGRkRAEodTXwYMHK33/3wyPsJvecCqh0vdJRFRZ\nmKRXIG9vb7z88suIjIxEbGwspkyZgqCgINy8eRNz5sxB48aNMXDgQOzYsQOyLCsdLhFRpZg5cyYO\nHDhg92rVqlWl79ddp4ZvsZtFlx+483C7RESOrkqSdKVbV5QQEhKCTz/9FHFxcVi/fj369u0LWZax\nfv169O7dG+Hh4VixYgVMJpPSoRIRVaimTZuiY8eOdi93d/cq2fdTbQLtpi/fzKmS/RIRVbQqbUlX\nqnVFSWq1Gv3798eWLVtw7tw5jB8/Hm5ubjh9+jReeOEFNG7cGF9//XWJp1AREdGDG9quHsRiA2xN\nWH0SJoukXEBERGVUpUm6kq0rjiA0NBQLFixAfHw8Zs6ciYCAAMTHx+Odd95BgwYN8PnnnyMjI0Pp\nMImIymX8+PFQq9Xw9PRE3759sXfv3irbtygK+PCJULuyzzadrbL9ExFVFI6TrgAfHx989NFHePvt\nt7FixQrMnj0bFy9exJQpUzBnzhy89dZbePvtt+Hr66t0qERE983LywsTJkxAjx494Ofnh4sXL2LO\nnDno0aMHfvvtN/Tt2/ee24iKioIk3V/Ld1F3QZPJhFOnTtnKBZMElaUA2UbrdvZdyMOqbXloWUtX\n6nac1Z3qj+4P66/8nKkORVFEcHDwA60jyFVwB2NkZCR69uyJ2rVrIzU1Fa6urujUqROmTJmCrl27\n2i0rSRKys7Ptyjw8PCCKldfoX/TEUY1Gg7Zt21bafu7EbDbjf//7H2bOnIm//voLgPUzFyXrfn5+\nVRqP0vWhuOPHgfbtgWPHgHbtWB+3YX3YK299KHHMux9Fx+37ceLECYSHh5c6LyMjA61bt4avr2+J\nH+HSPvvly5dhsZR/6MRso4TP9qTblbWtrcWIVu4Q+cA5IqpiKpUKjRo1siu717G+SpL0EydOYNmy\nZSVaV6Kjo0u0rpR20L569ep9t6yURfGbNzUazV2WrFySJGHnzp1YtGgRoqOjAQBubm54/vnnMWrU\nKHh6elZJHI5SH0pxOXcOzUaMQPSqVchv0cLp6+N2rA975a2P0lpXHCFJT0xMxG+//XZfyw4ePPiu\nV/5ef/11fP/998jLy4OLi4utvLzH+3vVfY5RwuozWTibcuvBRiFeGjzfxgt+rqr72kdNxv/L5cP6\nKz9nqsOyHOurJEkvzZ1aVyqzZaW6kCQJu3fvxg8//ICYmBgAgLu7O55//nkMHz7cqfrxK8H1/Hm0\nfOEFnF2+HHmhofdegagcytK6Ut289tprWLRoEfLz86HX623l5b2KcD9XMSySjJGLDyLPeOs3RBCA\nsCBPvNGrKep6u5S6njPgVbHyYf2VnzPVYVmOd4r1Sff29sbTTz+N77//Hvn5+XatK7dTq9WV+oPl\niGdyffr0Qe/evbFz50589913uHjxIhYtWoSff/4Zo0ePxnPPPXfXOisPR6yPqqRWq23vGo3G6evj\ndqwPexXRkl6TpaenY9OmTQgPD7dL0KuKShSw/G8d8Muxa/jtTAKy8s2QZeCv61l4bfkxNAvwwMMh\nPniqTSA89Pw+E5HjUPTG0aJGfOEe/QPDwsJqdJ/0u4mIiMDEiROxZs0aTJ8+HRcuXMA333yDVatW\n4ZNPPsErr7wCna5ib4Zy5PqoEoVXbZo1awa0bcv6uA3rw15l9EmvrkaOHIng4GA89NBD8Pf3R0xM\nDObNm4cbN25g6dKlisWlVYsY2SEY/cODsPFUAlYeumqbF30jG9E3srH2+DX0DK2NloGeaBrggUBP\nPUSRfdeJSDmKJelKt65UJ6IoYvjw4Rg6dCh++uknzJgxA7GxsXjrrbcwZ84cTJ48GWPGjGGrJhEp\nqk2bNvj555/x/fffIycnB76+vujatSuWL1+Ohx9+WOnw4K5TY8QjwXiogQ9+O5OI84nZuJ6RDwAo\nMEnYfCYJm88kAbC2wAd46hDo5YJ6Pi7o1zrQqbvGEFHVq5Ik3VFbV6obtVqNl156CSNGjMCSJUvw\n+eefIz4+Hq+++ipmz56NKVOmYNSoUbbuGkREVWnSpEmYNGmS0mHcU9MAD0wM8AAA3MgqwP+duI4/\no5Jgsty6RcsiyUjIKEBCRgGOxaVj/ckENA1wRz1vFwR6u8BTr4GHXg13vRoeOjU89Bq469Vw1ajY\nAk9EFaJKsjlHb12pbrRaLV599VW89NJL+OGHHzBz5kxcvnwZY8aMwcyZMzFt2jQMHz4cKhVHLyAi\nupsATz1e694Yozo2wIWkLFxMzsGlm7lIyMhHUmYBDOZbI83E3MhBzI2cu25PFABPFw383XXwddPC\nz10LfzcdPF3UcNWq4aZTWd+1arjqVHDVqqBXM7EnopKqJEmvLq0r1Y1er8dbb72Fv/3tb/juu+8w\ne/ZsxMTEYNSoUfj8888xefJkJutERPfBXadG+wa+aN/g1lCSsiwjJjkHqw5fxdXUPNzMMeBe46FJ\nMpCRZ0JGnunuCxYjCICLxpqwuxQm7XqtCi4aFfQasfDd+nLRFC6jEaFVqaBRCdBpVNCqRGjVIrQq\nETqNeGtaLUItCve894uIHA/7RdQAbm5ueO+99/Dqq69iwYIFmDNnDs6fP49Ro0bh008/xeTJkzFi\nxAh2gyEiegCCIKBZgAemPRMGADCYLUjIKEByVgFyjWZkF1hfOQYzsgtMyCmcTsszIj3XCOk+BziW\nZSDPaLEbJrIiiQKgKZa069QiNCoRuVmZUAkydBoVgq6dLUzyVbYkX6MWoSuW9BdtQ6cWoVYJUInW\nEwC1SoC62N8alQiVKBROF5aLAlQ8WSB6IMzaahAPDw989NFHGD9+PBYsWIB58+YhOjoaL774IqZN\nm4YPP/wQo0ePrvDRYIiInIFOrUJDfzc09He757IWSUZ6nhFpuUakZBuQYzDbEvE8oxm5hsJ3oxl5\nBgtyjWbkmyQUGC0oMFvu2WL/ICQZMJglu647AJCXb4IsyxAEM67mplXcDu9CJQrQFCX1KuFWMl84\nXZTYqwQBKpVgfS+W5N/6W4RKhN27WhQgFs0vWq9wH8WnRcG6P1EE1LevX7icKKDwvXC6cBuiCNu2\nco0SLBYJkiDBaJZs6/BEhCoKk/QayNPTEx9//DHefPNN/Otf/8JXX32F2NhYvPbaa/j000/x9ttv\n45VXXqmyJ5gSETkblSjA310Hf3cdmhXepHq/ZFmGwSyhwGRBgUlCntGMApOEfJMFBpMF+YUvk8Wa\nHBoLE3DjbdO3zy9eZjAr84BAiyTDIskAKu8p4lUlLz+v8CRHgOv+/bZyUbBehVEVT+yLJfwqQSic\nbz1JEAqTe+uy9uuohFtXI1SFJzCaonfVXcpE699F81TF5hfNs54wibYrIXZlvPLhEJik12AeHh6Y\nNGkS3nrrLSxevBhz5szB9evX8f777+Pzzz/H66+/jrfeeguBgYFKh0pERIUEQbD1Qa9MJ0+eRL7R\nBIhqNAttWSLJLz5tvG3aLMmwSNZ3s0WGSZJgscgwFZVb5MJlZJgst6bNtnlS4bxby1gk6/rS/fYT\nclCSDEAuOhmp3m6/8qG+Ldm3dXcqdqJQVKZVqaBVW5ct6mJVvNuVViUiPqkAkCxw0clQJ2Ra76lQ\nq6BRC7eWVVlfznhzNZN0J+Dq6ooJEybgtddew08//WTrsz5r1izMmzcPzz33HCZOnIj27dsrHSoR\nEVURQRCgEQVoNCL83B2nG6QkyZBka1Jve5duJf1mSYbFIsMiW08ILBJgliRIRe+y9cTBUmydor9L\nbs+6ftEJhyRb928pTLKL/r5VZr3SYZZkJCdbYLZYAEGEt69XKcvKkOWimG7brm37KLasfN/3MVSV\nyr7yYXc1IurMXZcVRQG6wsRdoxIK3+1vktaobr3rCu+f0GlU0KtF24mvrvDvohuzderCd03RDdmi\nw1xBYJLuRHQ6HcaOHYvRo0dj06ZNmDNnDvbu3YsVK1ZgxYoV6Nq1K8aNG4cmTZo4zBeUiIiciygK\nECFA7eADk9k/bbh1hWyzKFEvSo4tslx4hUK67aqEZHcVwlxsXtEJidkiwVT4XvxKhslSWFZ4FaRE\nWeHVDpPF/sqHufBKiblYDObCE5GqIEky8iVrV6/KJArW+090GhGiINjVj0WyP5H6qF8oOjX2q7Sc\niUm6ExJFEf3790f//v1x9OhRfPvtt1i9ejX27t2LvXv3wsfHBwMGDMDQoUP52HciIqIqIggCVIV9\n1KuDopOKohOFohMKk0WCySzf6iJlKSqTYCh8N1okXLx8BQaTBZIgonadILv7KEwW+/soTJZiXa4s\nsm0bRnPFtvJLMmz3fdzLF5vP48uhbdAisHLu8WOS7uQeeughLF++HLNnz8a///1vLFq0CAkJCVi6\ndCmWLl2Knj17YuzYsRgyZAhcXPhIbCIiIrK6dVJRtssep+Sbxa5GNCzTNoq6HxmL3yxd+F5gklBg\ntqDAZIHBZL1husBkvSk732QdbSk5qwDHr2aUad9atQhfN22Z1r0fTNIJABAUFISpU6fi448/xvz5\n87FmzRocOnQIO3fuxM6dOzFu3DgMGTIEo0aNQo8ePfiAJCIiIlKcIFhvbtWoxPtaXpJkfLs9BjvO\nJ9/X8h0a+sJVp4ZWZT/Sjk6twlOtA+HlqilP+HfFJJ3sqNVq9OrVC926dUNKSgoOHz6MJUuW4MqV\nK7bW9cDAQAwdOhRDhw5Fly5dmLATERFRpTJZJOQabj1ELNdohqGwVbxoyFJD4fCixaeLv+cZLLie\nkX/PfQ2MqIumtd0REewND33lJeH3wiSd7igwMBBTp07F5MmTsW/fPvz0009Ys2YNEhMTMX/+fMyf\nPx916tTBgAED0L9/f/Tq1Qt6vV7psImIiKgaS84uwKzN5xFzI6dK9zu2awg6NvJDoJdjdO9lkk73\nJIoiunXrhm7duuHbb7/F1q1b8csvv2DdunVISkrCokWLsGjRIri5uaFPnz7o168f+vbtiwYNGigd\nOhERETkoiyTjerYZlzON+D3hHAqMFhgtEv66nlUp+3PRqODpooGXiwYeejUMZgk5BjPCgjwx/OH6\n8HatvP7lZcEknR6ITqfD008/jaeffhpGoxHbt2/Hhg0bsGHDBiQkJGDdunVYt24dACA0NBSPPfYY\nevfujR49esDb21vh6ImIiKiq5BrMSMs1IjPfhIw8EzLyjdb3PCPiUvNwJu4mjBbJOk66S8WM0tK2\nvhde6hRya/zzwuEUq+MDkZikU5lptVr069cP/fr1w8KFC3H8+HFs3rwZW7ZswYEDB3D+/HmcP38e\nCxYsgCiKiIiIwKOPPopHH30UXbt2hb+/v9IfgYiIiMrAZJFsCXd64XtGngkpuQbEp+XjWnoeMvJM\nd9/GXZ7eJAp44Ic7DQgPwvMdGsBFWzPulWOSThVCEAS0b98e7du3x+TJk5Geno4dO3bYXufPn8ex\nY8dw7NgxfP311wCsLe2dO3dGly5d0LlzZzRr1gyieH93ZxMREVHVS8jIx/+OXsPOC8mFTyMtOz8X\nFYLcRTT11+OZzm3h62x9dI0AACAASURBVK6FtnAElerW6l0ZmKRTpfDx8cGQIUMwZMgQAMD169ex\nZ88e7Nq1C7t378bZs2dtLe0//vgjAMDb2xsdOnSwe/n5+Sn5MYiIiGo8SZJtI6Dkm6zjiheYJOSb\nLDCYLCgwW8cUPxKbdt9jinu7alDPxwW1PPTwLuwH7u1qfXm5aBHgqcPlC2dt46QH+7lW8qesfpik\nU5WoW7cuhg8fjuHDhwMAUlNTceDAAezbtw/79u3D0aNHkZGRgT/++AN//PGHbb3GjRvbJe3h4eHQ\n6XRKfQwiIqIaYdPpBCzadbnM67toVGhTzwu+7lp4u2jh62ZNvn3cNAjydoGngkMX1hRM0kkRfn5+\nthtQAcBkMuHMmTM4cOAADh06hEOHDiE6OhqXLl3CpUuXsHLlSgDWfvARERHo0KEDOnbsiM6dOyM4\nOBiCwMtiRI4iJycHkydPxpo1a5CWlobQ0FBMmjTJdpJORMr75di1Mq0X4KlDv1aB6BMWwES8kjFJ\nJ4eg0WjQrl07tGvXDuPHjwcApKen48iRI7ak/dChQ0hJSbH9/c9//hOAdTz3or7tXbt2RUREBNRq\nfrWJlDJ48GAcOXIEs2bNQrNmzbBy5UqMGDECkiRh5MiRSodH5LS2/JWIf+28dF/LDggPgl6jKnyJ\ncNGoUMtDh1ZBXuwvXkWYyZDD8vHxweOPP47HH38cACDLMi5fvoxDhw7h4MGDOHDgAE6ePInExESs\nXbsWa9euBYD/b+/O46Kq9/+Bv2ZfWRzABRAUkUVE9GpFBuYWYrlkRqJ9vVYPr7fUzPxdb3Zdr5ZZ\nXkrrcbvcL9kjS8VK7Vsqklu4peAKggsiCiga6wCzr78/BgZGEIZtFub9fDzmMTNnzpl5nw+HM+/5\nfD7n84FIJEJ0dDTGjh2LsWPH4oknnqAuMoTYSFpaGo4cOWJOzAFg3LhxKCoqwvLlyzFr1iyapZgQ\nO9DoDPhPhnUJevzQvpgfG9TNEZG22GwoDZlMhqVLl8LX1xd8Ph/Dhw/H7t27bfXxpAdgMBgYNGgQ\n5syZg88//xznz59HTU0NTpw4gY8++ghTpkyBp6cn5HI5jh07htWrVyM2Ntac7H/yySe4dOkSDIau\nGYuVENLcTz/9BLFYjISEBIvlr7/+OkpLS5GZmWmnyAhxbVw2EwO9RY99PayvG/q48zD36UC88cxA\nG0ZGHsdmNenU/Em6g1AoNI+9DgAGgwF5eXk4deoUMjIykJGRgfLychw5cgRHjhwBAHh7eyMuLg7x\n8fGIi4tDnz597LkLhPQoubm5CA8Pb9blbNiwYebXR48ebY/QCHF5m2YOQ0Ly2WbLXx7pj5A+bnAX\nsOHO50BnMACgFi97s0mSTs2fxFaYTCYiIyMRGRmJhQsXwmg04tq1azh27BiOHj2KjIwMVFRUYNeu\nXeaLUZ988klMnToVU6ZMQVRUFKinHSEdV1lZiaCg5s3kEonE/Hpr8vLyrG7t0mq15vvs7Ox2Rkqo\n/DrHWcuvH1+H29Uai2Xfns5vtp6fGxuTB4vh68aGB4/ZLQM0OGsZdgSTyURAQEC7trFJkt5a8+ec\nOXOQmZlJNSukWzAYDERERCAiIgJLliyBVqvFuXPnkJ6ejvT0dFy6dAlZWVnIysrC6tWrERAQgMWj\nR2M5AJ1ORxdtENIBrX2Zt/VFv+TQQ9QodY99PciTjRXP9LJYtulMNQqlj9+mwdTBQkwNaWzuV2oN\neOdw6z8aGrw32hODejWOZHGhVI3/vVzb5nZ8NgOfT7KcXfnbnDqcLlG1ue3Ivjz8daS7xbK/H6uE\nVNX2j5i5kWLEBgjMz+/X6fDPk9VtbgcAH4+XoJegseLsSKECP16Xt7mdr5iFdc9KLJZtyazBtQrN\nY7ZoNHGgAK8MEVssW3Cw3Kp4lzzhgaG9uebnuWUafH6+xqpt//cFH4vnP1yT4egdZZvbDfHmYulT\nHubnWq0W605UoVSmb3PbhHARngtqHBO8WqnHe8errIp37Zhe8HNr/GY6VazEd1dlbW7nyWfikwmm\neUdeDhPgp5sGnC5WQaF9/GREd6u1OFOshJjLQH93NnqLWBgTIECEDwfvHK6EStf2REYLRrhjlG/j\nNWG3q7X4+HfrxlnfGucFAaexV/b+fDn231K0uZ2jniM8BGz8318dMEmn5k/iKDgcDmJjYxEbG4sP\nP/wQpaWlSEtLw/79+3HkyBEUFxcjtbgYywEsnjAB/nFxCA8Lw+CQEPB5PEDf9gm4pxPk54Oj05n+\nn6k8Ol8eTCYwsOf0//Ty8mqxtryqypSENNSoP45UaUC18vFJqBvXgOI6U3Kg05m+dMvkBlS1sk2D\nB/LGbQFApTNatR0A3JcZwGE3bltq5WfyWLD4TAB4aOW2fygMzbatVBhQo7ZmX40W2z6UWb+vJTIj\n6pokYA/k1m3LYTKaxVum0Fu17UN58321Nt5SuRHudUaL59Zu29G/TZlCj8Lqxinv2Ww2yhXWHoeW\nfxupqj3HoRF6tP9vozc03VcmnhvkhiKp3qofUFqDEVK1EVK1DvlVdQCAP2R66K2YbLT0kb/rfZl1\nZQSYjkM+u+m+Wrdt03NEA+v/Nt13jjCi/dfDMYxGY+fmdLVCSEgIgoKCkJ6ebrH8wYMH8PX1xcaN\nG/H+++8DMPUprqurs1ivuLi4Wy/2a2huAUxJnKtz1fJQKpU4d+4crv/yC/7722/2Doe4CIOXF+pu\nW4644ObmBibTZtf1d6kFCxYgNTUV1dXVFhUzu3fvxuzZs3HmzBlzpUxL5/uxW86hupWadEIIcUa9\nBGxkLI22WNbWud5mrfmdaf7U6XTQ26jGrmmCSlyrPNhsNmJiYhATE4Mrd+7g5sWLyMrMxIULF1An\na2xOlPTqhejoaDzzzDM0kRLpNCaXi372DqILzZgxAykpKdi7dy9mzZplXr59+3b4+vriqaeeanX7\nnfOjra5vys/PN3VLY7MREhLSiahdE5Vf57hK+VXK1diVWYzbZW13ebIGgwHwOSwIOCzo1QpwmUbw\nOUz49/WBgMOCgMuCkMs2PeaxIOKwIOCyIeCatmE56RjtHal2sUmS3tnmTzab3a21Sq5ac/w4VB4A\nQkIQFhKCsNmzkaBU4sKFCzh8+DB+++03yKqrcezQIeDQIQwaNAgvvPACnn/+efTt29feUdsEHR+W\nOlsezlpj/jiTJ0/Gc889h7feegu1tbUIDg5Gamoq0tPTsWPHjjYHCQjrZ30rgr6CC62WAQ6Hg6F+\nHm1vQCxQ+XWOK5XfsyG9zY/1BiMUGh3kaj1kah0UGh1kah1kKh2kCi2qFBoUVcqRe7/1azaUWj0U\naj2MRiMYDAMqtNZdRyDgsCDmm5J2DwEH854egNC+bp3aP1toqeWwLTZJ0iMjI5Gammr+xdng6tWr\nAIChQ4e2un1ERES3fpFlZ2dDq9WCw+EgKiqq2z7HWVB5WMrOzkZ0dDRiY2MRGhqKQ4cOYdeuXdi/\nfz9u376Nzz//HF988QXGjx+PP//5z3jppZcgFovbfmMnRceHpc6WR0dO3I5u3759WLlyJdasWYOq\nqiqEhYUhNTUViYmJ9g6NENJJLCYDbnwO3PitV0oUlsuQdacKMrUOap0BcnVjMi+rf6xSAu3tJ6HU\n6qHUNm71tx+zMS6sN3zEXHiLefAS8+At5sLXUwA+x7lHDrRJkt7Z5k9CHAWfz8eMGTMwY8YMSKVS\n7NmzBzt27MCJEydw7NgxHDt2DAsXLsTLL7+M1157DWPGjOlxNaWEtEUsFmPr1q3YunWrvUMhhNhJ\nkI8YQT6tV1hduXIFcpUWWrAQMCgEcrUedWot5Go95God6tQ6U3Jfn9g3JPr3qi1H4PntRlmz92Yw\ngH4efAzwEuGZYG/EDvZ2uu6pNknSO9v8SYgj8vT0xPz58zF//nzcvXsXO3bswPbt21FQUIDt27dj\n+/btCAwMxLx58zBv3rwWx44mhBBCXBWDwQCPzYCYw2ozoW/KaDQi42Y5Dl97iGultTC0MASK0QiU\nSlUolarw++1KXCyqxrvPOde1Aza7cJSaP0lPNmDAAKxatQorV67E2bNnsX37duzevRtFRUVYv349\n1q9fjzFjxmDevHlISEiAm5vj958jhBBCbMFoNEKp0UOuMdWWy9V6c193haa+77taB7nGVMOuaHIv\n4LIgV7fdaeZKiXXjszsSmyXp1PxJXAGDwcDo0aMxevRobNmyBT///DO++eYbHD58GCdPnsTJkyex\nePFivPTSS5g3bx7Gjx9PLUmEEEKcht5ghEZngFqnh1pnqH9sutfoDVBr9fX3pucqrR5yjf6RJNuU\niD+oqIJSo4daD/BPne3SOPkcJsQ8NipkGowM7IVpw3279P1tgSZUJKSbCAQCJCYmIjExEffu3TN3\nh7lx4wZ27tyJnTt3wtfXF6+++irmzp2LyMhIe4dMCCGkh9DWJ8hKrR4qjQEqnR5Kjb5xmdaUZCs1\nDc8NUGr1UNe/rtToodKZ3kOl1UOtNUCtN8DQUt+SDlIoG0Z36VhfcT6HCRGPDRGXDSGXhQHeIrw8\n0h+eQg54bOevAKMknRAb8Pf3x4oVK/Dee+8hKyvL3B2mtLQUmzdvxubNmzFs2DC8+uqrmD17Nvr3\n72/vkAkhhNiB0Wisr4E21UqrtKZaa5XWlGjL1DrUKrWmm0qHGqUWNQot6tRaKMxJeNcm092BxWRA\nzGGCxwKEXBb8+3rUJ9tsiHgsiHimxFvEZZsS8fplovrXhVy2046Zbi1K0gmxIQaDgaeeegpPPfUU\nPvvsM6SlpeHbb7/FwYMHkZOTg5ycHLz33nuIjY3FrFmz8PLLL6NPnz72DpsQQkgTBoOxflhBAwxa\nHe5UyM010w01zw3Jtbppkq3VQ6V7JPFuYV1Hya85LAb4HBb4HBZ4bCa4bCa4LCZ4HCa4LFb9vWk5\nr/7GZTc8Z5nXNSfeTRJuLouJnJycJkPYUmvyoyhJJ8ROeDyeeTjHqqoq7NmzBzt37sTJkydx6tQp\nnDp1CkuWLMHYsWMxc+ZMzJgxA/369aS5KQkhxHHoDUbzhYsFZTKcvV2JaoUG6voa7MaEWg+t3giF\nUmHuqiG8cNne4UPAYUHIMyXTDTN38timez6bBT6H2fi4fvZOPodpnv2z4XHjc+ed3bOnoCSdEAcg\nkUiwYMECLFiwACUlJfjxxx+xe/dunD9/HsePH8fx48exePFijB49Gi+++CKmTZvWo6ehJoSQjtDo\nDCiXqaFQW44MIm+41V+42LjMNKKIQm05QY4tcdlMU4LMbqyx5jVNmtmmGms+2/RcyGPDQ8CBO7/+\nXsCBO58DLpvm5OhpKEknxMH0798fy5Ytw7Jly1BYWIi9e/di7969yMzMxJkzZ3DmzBksX74coaGh\nmDJlCiZPnoyYmBjweDx7h04IId1KqzegWqFBjUILqVILqUKLaoUGUoUGBWUy3CqTQafv2r4iTAbA\na0iU62uY+WwmZDU6sBhG8DksBPr1qX+N2eK65qS7PhE3r8dmgkm11eQxKEknxIEFBQVh+fLlWL58\nOUpKSvDzzz9j//79+O2333Dz5k3cvHkTSUlJEAqFGDduHCZOnIgJEyYgIiKCZjolhDgllVaPhzUq\nPKhR4UGNsvFeqkKFTN2l/bV5bNPoIOKGixTrH3sKOXhyoAShfd3AZTFbHH0kOzu7SX/qwV0XFCH1\nKEknxEn0798fixcvxuLFi1FbW4tff/0VaWlpSE9Px8OHD3Hw4EEcPHgQAODj44MxY8YgNjYWMTEx\niIqKAptN/+6EENvT6g2oVWpNo5A0uTVdJlVoUasyPbZmYprH6efBR3BvMdwFHIjqk+6GxNs0Mkjj\nCCFCHgscFlVmEMdF39qEOCF3d3ckJCQgISEBRqMROTk5OHz4MI4fP46TJ0+ivLzc3E0GAEQiEUaO\nHGkeWWbkyJEIDAzs8Ni0hBDSkuJKBY7d+ANldWr8UavCA6kKMrWuS95byGXBz1MAHzcePIQceAq4\n8BRy4CngwFPIha8nH55Cbpd8FiGOgJJ0Qpwcg8FAVFQUoqKisHz5cmg0GmRlZZlHiDlz5gxqa2vN\nM5426NWrF0aMGIGoqChERkYiMjISQ4YMgVAotOPeEEKciUZngFRp6iNeXKXAlqO3OvxeQi4LHgIO\nPIUc9HXno5+nAP08+PD1FKCvBx9uPDZVLBCXQkk6IT0Ml8tFTEwMYmJi8P7770Ov1+PGjRvIzMxE\nVlYWsrKykJubi+rqavPIMQ0YDAYCAwMRHh6OsLAwhIaGIiQkBCEhIfD19aUvSEJcmNFoxM0/6vB/\nl0tRWC6DVKmFUtN615TebjxIRFx4CDimm5BjHpHEQ2CqBafRSQhpGSXphPRwLBYLERERiIiIwBtv\nvAEA0Gg0yMvLw6VLl5CTk4Pc3FxcvXoV5eXluHv3Lu7evYtDhw5ZvI9QKMSgQYPg4+MDPz8/BAYG\noqKiAoMGDUL//v3BYjn/FMyEEEt6gxHXSmtxrrAS5worUVantnrbddOGYGSgpBujI6RnoySdEBfE\n5XIxYsQIjBgxwmJ5eXk5bty4gevXr+P69evIz8/HrVu3UFhYCIVCgatXr1qs/8EHHwAAOBwOBg4c\niODgYAQHByMkJMRcC+/v708jzRDiwIxGI+5WyFGl0KBarkG1QgupQoNymRpX79WgTtW8TzmHxYCP\nG8/cL7yhj7i3mIshvu7w8xRQyxshnURJOiHEzMfHBz4+PoiNjbVYrtFoUFRUhIKCApw4cQJ3795F\naWkpysvLUVhYCI1Gg/z8fOTn5zd7T6FQiPDwcISHhyMiIgLDhg1DVFQUdZ8hxI5UWj3uVmtQJFXj\n7D0NZMa2Z8xkMhmI8vdATLA3xob2pu4phHQzStIJIW3icrkYPHgwBg8eDF9f3yZjA0dBr9fj/v37\nKCgoQEFBgbn2PT8/H7dv34ZCocDFixdx8eJFi/f08vLCyJEjMWrUKIwaNQrR0dHo16+fnfaQENdR\nXKnA+z/l4GFVbeO09oKWR0URcFj4U2AvRAdJMGqABGIepQ2E2Ar9txFCOoXFYiEgIAABAQEYP368\nxWs6nQ63b9/G9evXkZeXh7y8PGRnZ+PmzZuorKzE4cOHcfjwYfP6AwcONF/0OmHCBAQFBVFtOyEd\nJFVocKagEg9qlBbjkd+pkLe4/vThvujrwUcvIdd0E3HgLebRWOKE2Akl6YSQbsNmsxEaGorQ0FC8\n+OKL5uUqlQq5ubm4ePEiLly4YB5x5s6dO7hz5w6+++47AEBgYCAmTpyI+Ph4xMXFwd3d3V67QohT\nuHqvBmduVyD/YR1ul8usnp3zi9kjMMBb1L3BEULahZJ0QojN8fl8czeXv/71rwCA2tpanDt3DqdP\nn0ZGRgbOnTuHoqIibNu2Ddu2bQOHw8Gzzz6LqVOnYubMmfDz87PzXpCWZGRkYNy4cS2+dvbsWURH\nR9s4Itdx82Ed/vHT1VbXYTIZ5qEPodLDT8xEdICIEnRCHBAl6YQQh+Du7o64uDjExcUBAORyOU6d\nOoUjR47gwIEDyM/Px9GjR3H06FEsXboUzzzzDF555RXMmjULvXv3tnP05FEbN25slqwPHTrUTtG4\nhiq5ptmyhFH++FNAL9MILAIOxE0mBMrOzq6/voSGTyXEEVGSTghxSCKRCPHx8YiPj0dSUhLy8/Nx\n4MAB7N27F7///jtOnz6N06dP491338XkyZMxb948TJ06FTwez96hEwCDBw+mWvNuVlQpR1GlAlVy\nDSpkavx8pdTi9bC+bvjz0wPsExwhpNNskqRT8ychpLNCQkKwbNkyLFu2DPfu3cOePXuQmpqKrKws\nHDhwAAcOHIC3tzdef/11LFiwAMHBwfYOmZAuZTAYUVAuw32pEt+cudtizXlTz0fSaEmEODObXrK9\nceNGnD171uJGzZ+EkPby9/fH0qVLkZmZiWvXrmHFihXw8/NDRUUFNm/ejMGDB2PSpElIS0uDwWCw\nd7guadGiRWCz2XB3d8ekSZNw+vRpe4fk9DamXcf/+yEbnx7ObzVBF/PY2PPW0xgXRt3ACHFmNu3u\nQs2fhJCuFh4ejo8++ggbNmxAWloakpOTkZ6ebh7eMSwsDO+88w7mzZsHgUBg73B7PA8PD7zzzjsY\nO3YsvLy8UFBQgM2bN2Ps2LE4ePAgJk2a1Or2eXl5Vv+w0mq15vvs7OxOx+4oDEYj8srUuF6hQZ3a\nAJnGgDqNAVVKfYvrP+knwBO+fLjzmPDgs8BlMXAjL7fNz+mp5WcrVH6d50plyGQyERAQ0K5tqE86\nIaRHYLPZmDZtGqZNm4bCwkL8+9//xldffYUbN27grbfewtq1a7F06VIsXLgQHh4e9g63xxoxYgRG\njBhhfh4bG4sZM2YgMjISf//739tM0nU6HfT6lpPR1jR82Tubgiot8so1qFUbUKs2oKb+3pqfKe8+\n6QEvoSkpNzECBh20HWg8ctbycxRUfp3X08uQxWr/Bdo2TdIXLVqExMRECIVCPP3001i9ejViYmJs\nGQIhxAUEBQUhKSkJa9euxddff40tW7agqKgI//jHP7Bp0ya8/fbbWLZsGSQSib1DdQmenp6YMmUK\nkpOToVQqW23RYLPZYDKt64nZ9Eudw+F0Os7upNEbUa3Uo0qpR7VKj3K5HtcrNCiT65qvzGDg0Sm8\nRBwmxFwmDEYjAjw4GD9QhH5unfsKd6byc0RUfp3nSmVo7XmtKYbRaLRyqoOOu3z5MrZv396s+TM/\nP79Z86fBYEBdXZ3F9sXFxd3ar9SVDhJrUHlYovKw5IzlodVq8euvv+Lrr79GYWEhANPoMXPmzMHc\nuXM7NUlSZ8ujpSZQNze3Dp3QHdmbb76J//73v1AqleDz+QBaPt+3Z98bhxDkICoqqstjtpbRaES5\nTI2SKgXuS1WolmtQJdegWtF4X6tsIRlvgYeAA4mIiz7uPIwL7Y2wfu7wEHDAYnb9zLuOUn7Oisqv\n81ypDDtyvmt3kt7aSC2Punz5MoYPH97ia1KpFJGRkZBIJBb9kFraicLCwg41fxJCSFMGgwEnTpxA\nSkoKbt26BcCUrM+dOxezZ8+GUCi0eUwsFgtBQUEWy3pakl5dXY3IyEj4+Pjg8uXL5uXOlqQbjUZU\nyTUoqlKgpEqBokoFiqsUKK5UQKnt2HeUh4CDvh58vPlsEAK9ROCwbPd3d6UEqTtQ+XWeK5VhR853\n7W4rCw0NRUpKilXrttZBvruaPzvCGWsGuxOVhyUqD0vOXh5xcXGYOHEijh8/juTkZBQUFCA5ORk/\n/PADFixYgJkzZ7Zrv7qiJr0nmTNnDgICAjBq1Ch4e3vj1q1bSEpKwh9//IFvvvnG3uFZTas34ObD\nOtypkKO4SoGiStO9XN2+ZJzNYkAi5MJbzENvdx56u/Hg48ZHXw8+BngJ4SnkdtMeEEKcXbuT9H79\n+mH+/Pld8uENlfgNs589TkRERLd+kbnSLzlrUHlYovKw1FPKY8SIEXj33Xfxww8/YNWqVbh9+zY2\nbdqE77//Hh999BFeeeWVNs9NQOfLo6XaFWc2bNgwfP/990hOToZMJoNEIkFMTAy+++47PPHEE/YO\nr0WlUiXuVspRVqvGw1oVHtaocK201ura8T7uPPSXCBEoEcK/lxDebjxIhFz0ElnO8EkIIe1ht9Fd\nqqurceDAAQwfPtzcP5EQQmyJyWQiMTERM2fOxLZt27B+/XrcuXMHiYmJ+PTTT/Gvf/0LsbGx9g7T\nqaxYsQIrVqywdxhWqZCpsf33u8i4WW7V+t5iLgK9ROgvESJAIsQAL1NSLuC2f9QGQghpi02S9J7S\n/EkI6Zk4HA7efPNNzJ07F0lJSfjkk0+QlZWFMWPG4KWXXsLHH39MM5g6OaVGj+x7UlwulqKoUo6S\nakWrF3O6C9gYGdALQ3w9EOhlSspFPBq1mBBiOzY54zhj8ychxPWIRCKsWbMGCxYswLp165CSkoJ9\n+/Zh//79ePvtt7F69Wp4enraO0zSTrn3a/D+vquPfV3MY+P5Yf3g7ylAH3c++rjz0EvIBbMbRlQh\nhBBr2eSKpRUrVuDy5cuQSqXQ6XQoKyvDvn37KEEnhDikvn37Ijk5GTk5OZg0aRK0Wi0+/fRTBAcH\n4z//+Q90OuuG0yP2pdMb8OOFkhYT9F4iLob5e2Dmn/yQ/D8jMTc6EOPCemOIrzu8xDxK0Akhdkdt\nd4QQ8hgRERFIT09Heno6li1bhuvXr2PhwoX48ssvsWXLFkyYMMHeIZLHuFIixadH8lEt11gsnx87\nEBPC+0BMXVcIIQ6uZ439RQgh3SA+Ph7Z2dn44osvIJFIkJubi4kTJ2L69OkoKiqyd3jkETK1Dh8c\nuGaRoI8L9cG2eaMwfbgfJeiEEKdASTohhFiBw+Fg8eLFuHXrFpYsWQIWi4VffvkFM2fOxNatW3vU\nMIrOrlqugVrXOEv1hheHYllcKHq700hihBDnQUk6IYS0g0QiwdatW3H16lXEx8dDp9Nhx44dmDZt\nGpKTk6m/up0pNXr8UauyWPbDhRKUSpV2iogQQjrG4dr8GiY4aspgMLSwZtdhMplgsVhgMpnd/lnO\ngMrDEpWHJSoPk9DQUBw8eBDbtm3Dl19+iZKSEqxatQo7d+7E+vXr8eyzz1r1Pi2VYUvnwZ6os+d7\nJpOJMqURp/PrsDP/CirkGqg0pgmI3HiNY5ffLZfhaN5D/M/TgZ0Pugeh/+XOofLrPFcqw46c6xlG\nB/s20Ol0kMvl9g6DEELsRiQSgc12uDqULkfne0KIK2vrXE/dXQghhBBCCHEwlKQTQgghhBDiYChJ\nJ4QQQgghxME4XJ90g8HQrHM9g8EAg0GzvxFCeh6j0djs4iEmkwkms+fXodD5nhDiKjpyrne4JJ0Q\nQgghhBBX1/OragghhBBCCHEylKS34auvvgKDwYBYLLZ3KHZx/PhxvPHGGwgLC4NIJIKfnx+mT5+O\nixcv2ju0bieT8K4GnAAABChJREFUybB06VL4+vqCz+dj+PDh2L17t73DsgtXPg6s4ernCUIIIV2P\nuru04v79+4iIiIBIJEJNTQ1kMpm9Q7K5hIQEVFZWIiEhAUOGDEF5eTmSkpJw4cIF/Prrrxg/fry9\nQ+w2cXFxOH/+PDZt2oSQkBDs2rULX331FXbu3Ik5c+bYOzybcuXjoC10niCEENIdKElvxdSpU8Fg\nMCCRSLBnzx6X/PItKytD7969LZbJZDIEBwdj6NChOHr0qJ0i615paWl44YUXsGvXLsyePdu8PC4u\nDnl5eSguLgaLxWrlHXoWVz0OrEHnCUIIId2Burs8xo4dO3DixAl8+eWX9g7Frh5NzABALBZjyJAh\nKCkpsUNEtvHTTz9BLBYjISHBYvnrr7+O0tJSZGZm2iky+3DV46AtdJ4ghBDSXShJb0FZWRmWLl2K\nTZs2wd/f397hOJyamhpcunQJERER9g6l2+Tm5iI8PLzZdL3Dhg0zv+7qXOE4aA2dJwghhHQnStJb\nsHDhQoSGhuKtt96ydygOadGiRZDL5Vi5cqW9Q+k2lZWVkEgkzZY3LKusrLR1SA7HFY6D1tB5ghBC\nSHfq0Ul6RkaGeWKMtm5XrlwBAOzduxf79+9HSkpKj5tQoyPl8ajVq1dj586d+OyzzzBy5Egb74Ft\ntfb372nHRnu50nHQkp58niCEEOIY2G2v4rxCQ0ORkpJi1boBAQGQyWRYtGgR3n77bfj6+kIqlQIA\nNBoNAEAqlYLD4UAkEnVbzN2pveXxqH/+85/44IMP8OGHH2Lx4sVdHZ5D8fLyarG2vKqqCgBarGV3\nFa50HLSkp58nCCGEOAgjMbtz544RQKu36dOn2ztMu1i3bp0RgHHdunX2DsUm/vKXvxjFYrFRq9Va\nLE9NTTUCMJ45c8ZOkdmXqx0HLaHzBCGEEFugIRibUKlUOHfuXLPlmzZtwokTJ3Do0CF4e3tj6NCh\ndojOfjZs2IA1a9Zg1apV2LBhg73DsYlDhw7h+eefx+7duzFr1izz8smTJyMnJ8flhmAEXPM4aAmd\nJwghhNgCJelWeO2111x2/OOkpCT87W9/Q3x8PNauXdvs9ejoaDtEZRtxcXG4cOECPv74YwQHByM1\nNRUpKSnYsWMHXn31VXuHZ1OufBxYy5XPE4QQQrpej+6TTjpv//79AID09HSkp6c3e70n/8bbt28f\nVq5ciTVr1qCqqgphYWFITU1FYmKivUOzOVc+DgghhBB7oJp0QgghhBBCHEyPHoKREEIIIYQQZ0RJ\nOiGEEEIIIQ6GknRCCCGEEEIcDCXphBBCCCGEOBhK0gkhhBBCCHEwlKQTQgghhBDiYChJJ4QQQggh\nxMFQkk4IIYQQQoiDoSSdEEIIIYQQB0NJOiGEEEIIIQ6GknRCCCGEEEIcDCXphBBCCCGEOJj/D9JE\n9X2TNnlyAAAAAElFTkSuQmCC\n", "text/plain": [ "<matplotlib.figure.Figure at 0x1bcf331c6d8>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "y = g2(data)\n", "gaussian2 = (np.mean(y), np.var(y))\n", "\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": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "input mean, variance: -0.0018, 1.0010\n", "output mean, variance: -0.1223, 2.4067\n" ] } ], "source": [ "print('input mean, variance: %.4f, %.4f' % \n", " (np.mean(data), np.var(data)))\n", "print('output mean, variance: %.4f, %.4f' % \n", " (np.mean(y), np.var(y)))" ] }, { "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": {}, "outputs": [ { "data": { "image/png": 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nY8aMwWw289lnn7Fz585zrvff//6XY8eO0a5dO2dXmjPdLSu7CrBl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\n6WKe8Sjp54pr3PZ/T9yGSmbGEUL4ABnJF0KIFtKlSxeeeOIJtmzZQn5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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf3279be0>" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "output mean, variance: 0.0026, 2.2522\n" ] } ], "source": [ "def g3(x): \n", " return -1.5 * x\n", "\n", "plot_nonlinear_func(data, g3, gaussian)\n", "out = g3(data)\n", "print('output mean, variance: %.4f, %.4f' % \n", " (np.mean(out), np.var(out)))" ] }, { "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": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "linear output mean, variance: 0.1520, 7489.2679\n", "nonlinear output mean, variance: -8.8097, 30501.0215\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 the nonlinear version is not stable. It drifted significantly from the mean of 0, and the variance is half an order of magnitude larger.\n", "\n", "I minimized the issue by using a function that is quite close to a straight line. What happens if the function is $y(x)=x^2$?" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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jsFKDw5cLcTm/ssn1ege5Y8m4aA77RmSmXFVKPNk3EE/2DURhRQ3yymuQml+B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\nNFi9ejUGDBiAy5cvix2NGqlfv344ePAgwsPD4ePjg8jISDg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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf0fee780>" ] }, "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", "plot_nonlinear_func(data, g3, gaussian=x0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Despite the curve being 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've tried to track a ball 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 hard to look at probability distributions and reason about what will happen in a filter. So let's think about tracking an aircraft with radar. The estimate may have a covariance that looks like this:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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yIkM3HG9iJFCgVtbRrJZYj05EGPkJTq2FUCQV+6sHYGsltOs2lloVrqbT1GNY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\n", "text/plain": [ "<matplotlib.figure.Figure at 0x1bcf103b7f0>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import kf_book.nonlinear_internal as nonlinear_internal\n", "\n", "nonlinear_internal.plot1()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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YRMMKc2qiRMOKlLoIcQal4gH+5t1L+cNTHfzh6WMwWPM61+W+jR3s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\ni0xsZwYhxPSVKxr0pDUsG2oiNdREagk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CyGmomp4J6aEUdJ2B2+yB2+KGwPLwuSyodNsg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\nLesGUbtFAstS/zkhpLiYJQGfOL8Rf94+MkOkFRWb3z+GGy+dYVBlpYX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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf11a8400>" ] }, "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 Extended Kalman filter 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": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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o3uh2JIv85LeH+OVjR7h0RS1Xrq6noUaGSMwkqmaRLxk0R+vxeWUIzGwjYV0I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A/c//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\n9f8BdAbAzdpoezYAAAAASUVORK5CYII=\n", "text/plain": [ "<matplotlib.figure.Figure at 0x1bcf1219470>" ] }, "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": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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YYNJ7pFPxIBetauSiVY0YpsOOfXlefT3Dph0ZTGt049x10+H5zT08v7mHmmSI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"text/plain": [ "<matplotlib.figure.Figure at 0x1bcf12e4320>" ] }, "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", "This covariance ellipse 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 will still accumulate. Other systems you may be work could have this amount of nonlinearity - this was not an exaggeration only to make a point. You will always be battling divergence when working with nonlinear systems." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Algorithms" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You may be impatient to solve a specific problem, and wondering which filter to use. I will quickly survey 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* (EKF). 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 more 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 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, the algorithm is robust even for extremely nonlinear problems. 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 good problem for GPUs because the part of the algorithm is parallelizable. Another cost is that 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 and standard deviations 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). It does not require you to find analytic solutions to nonlinear equations, and yet almost always performs better than the EKF. It does well with nonlinear problems - problems where the EKF has significant difficulties. Designing the filter is extremely easy. Some will say the jury is still out on the UKF, but to my mind the UKF is superior in almost every way to the EKF. I suggest that the UKF should be the starting point for any implementation, especially if you are not a Kalman filter professional with a graduate degree in control theory. 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. 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 filter as your starting point. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Summary" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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 to 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. 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. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "<A name=\"[1]\">[1]</A> https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb" ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.3" } }, "nbformat": 4, "nbformat_minor": 1 }