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      "The Extended Kalman Filter"
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     "source": [
      "**Author's note: this is still being heavily edited - there is a lot of duplicate material, incomplete material, and so on. In particular, there is a lot of blathering on about linearizing a function at a point. There is not a lot of content there - please feel free to skip ahead if you get the idea - use the slope at the point as the linearization of a function.**\n",
      "\n",
      "The Kalman filter that we have developed to this point is extremely good, but it is also limited. Its derivation is in the linear space, and hence it only works for linear problems. Let's be a bit more rigorous here. You can, and we have in this book, apply the Kalman filter to nonlinear problems. For example, in the g-h filter chapter we explored using a g-h filter in a problem with constant acceleration. It 'worked', in that it remained numerically stable and the filtered output did track the input, but there was always a lag. It is easy to prove that there will always be a lag when $\\mathbf{\\ddot{x}}>0$. The filter no longer produces an optimal result. If we make our time step arbitrarily small we can still handle many problems, but typically we are using Kalman filters with physical sensors and solving real-time problems. Either fast enough sensors do not exist, are prohibitively expensive, or the computation time required is excessive. It is not a workable solution.\n",
      "\n",
      "The early adopters of Kalman filters were the radar people, and this fact was not lost on them. Radar is inherently nonlinear. Radars measure the slant range to an object, and we are typically interested in the aircraft's position over the ground. We invoke Pythagoras and get the nonlinear equation:\n",
      "$$x=\\sqrt{slant^2 - altitude^2}$$\n",
      "\n",
      "So shortly after the Kalman filter was enthusiastically taken up by the radar industry people began working on how to extend the Kalman filter into nonlinear problems. It is still an area of ongoing research, and in the Unscented Kalman filter chapter we will implement a powerful, recent result of that research. But in this chapter we will cover the most common form, the Extended Kalman filter, or EKF. Today, most real world \"Kalman filters\" are actually EKFs. The Kalman filter in your car's and phone's GPS is an EKF, for example. "
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     "source": [
      "The Problem with Nonlinearity"
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     "source": [
      "You may not realize it, but the only math you really know how to do is linear math. Equations of the form \n",
      "$$ A\\mathbf{x}=\\mathbf{b}$$.\n",
      "\n",
      "That may strike you as hyperbole. After all, in this book we have integrated a polynomial to get distance from velocity and time:\n",
      " We know how to integrate a polynomial, for example, and so we are able to find the closed form equation for distance given velocity and time:\n",
      "$$\\int{(vt+v_0)}\\,dt = \\frac{a}{2}t^2+v_0t+d_0$$\n",
      "\n",
      "That's nonlinear. But it is also a very special form. You spent a lot of time, probably at least a year, learning how to integrate various terms, and you still can not integrate some arbitrary equation - no one can. We don't know how. If you took freshman Physics you perhaps remember homework involving sliding frictionless blocks on a plane and other toy problems. At the end of the course you were almost entirely unequipped to solve real world problems because the real world is nonlinear, and you were taught linear, closed forms of equations. It made the math tractable, but mostly useless. \n",
      "\n",
      "The mathematics of the Kalman filter is beautiful in part due to the Gaussian equation being so special. It is nonlinear, but when we add and multipy it using linear algebra we get another Gaussian equation as a result. That is very rare. $\\sin{x}*\\sin{y}$ does not yield a $\\sin(\\cdot)$ as an output.\n",
      "\n",
      "> If you are not well versed in signals and systems there is a perhaps startling fact that you should be aware of. A linear system is defined as a system whose output is linearly proportional to the sum of all its inputs. A consequence of this is that to be linear if the input is zero than the output must also be zero. Consider an audio amp - if a sing into a microphone, and you start talking, the output should be the sum of our voices (input) scaled by the amplifier gain. But if amplifier outputs a nonzero signal for a zero input the additive relationship no longer holds. This is because you can say $amp(roger) = amp(roger + 0)$ This clearly should give the same output, but if amp(0) is nonzero, then\n",
      "\n",
      "> $$\n",
      "\\begin{aligned}\n",
      "amp(roger) &= amp(roger + 0) \\\\\n",
      "&= amp(roger) + amp(0) \\\\\n",
      "&= amp(roger) + non\\_zero\\_value\n",
      "\\end{aligned}\n",
      "$$\n",
      "\n",
      ">which is clearly nonsense. Hence, an apparently linear equation such as\n",
      "$$L(f(t)) = f(t) + 1$$\n",
      "\n",
      ">is not linear because $L(0) = 1$! Be careful!"
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     "source": [
      "The Effect of Nonlinear Transfer Functions on Gaussians"
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     "source": [
      "Unfortunately Gaussians are not closed under an arbitrary nonlinear function. Recall the equations of the Kalman filter - at each step of its evolution we do things like pass the covariances through our process function to get the new covariance at time $k$. Our process function was always linear, so the output was always another Gaussian.  Let's look at that on a graph. I will take an arbitrary Gaussian and pass it through the function $f(x) = 2x + 1$ and plot the result. We know how to do this analytically, but lets do this with sampling. I will generate 500,000 points on the Gaussian curve, pass it through the function, and then plot the results. I will do it this way because the next example will be nonlinear, and we will have no way to compute this analytically."
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      "import numpy as np\n",
      "from numpy.random import normal\n",
      "\n",
      "data = normal(loc=0.0, scale=1, size=500000)\n",
      "ys = 2*data + 1\n",
      "\n",
      "plt.hist(ys,1000)\n",
      "plt.show()"
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Kqq4BACZBiAZQciZb5mAYxjz3ZG7YbabiiaT6B0O6bEGd1d0BAOTAmmgAKDITy915/OGy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\nozMQPr8eggAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5f0343e48>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is an unsuprising result. The result of passing the Gaussian through $f(x)=2x+1$ is another Gaussian centered around 1. Let's look at the input, transfer function, and output at once."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from nonlinear_plots import plot_transfer_func\n",
      "\n",
      "def g(x):\n",
      "    return 2*x+1\n",
      "\n",
      "plot_transfer_func (data, g, lims=(-10,10), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dB4/x48ELFb6JbPr1z9VFpg5qi2GRLYwTmIjIBFbuPo8Z3xyBIAAju4dgzriu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\ndI1Qbv+egwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5efc73978>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The plot labelled 'input' is the histogram of the original data. This is passed through the transfer function $f(x)=2x+1$ which is displayed in the chart to the upper right. The red lines shows how one value, $x=0$ is passed through the function. Each value from input is passed through in the same way to the output function on the left. The output looks like a Gaussian, and is in fact a Gaussian. We can see that it is altered -the variance in the output is larger than the variance in the input, and the mean has been shifted from 0 to 1, which is what we would expect given the transfer function $f(x)=2x+1$ The $2x$ affects the variance, and the $+1$ shifts the mean.\n",
      "\n",
      "Now let's look at a nonlinear function and see how it affects the probability distribution."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from nonlinear_plots import plot_transfer_func\n",
      "\n",
      "def g(x):\n",
      "    return (np.cos(4*(x/2+0.7)))*np.sin(0.3*x)-1.6*x\n",
      "\n",
      "plot_transfer_func (data, g, lims=(-4,4), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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vjEgPsQEm0lG2FoUXyLe3MQMASAtKH7Nsb2WKNZM7q7XPOZv+wNX40hvt0pQ0\nvjotKxefDm/F65wSVYDMUIqB7epgQFtv/Hn9ETb8fhlHzt3FLyeu4JcTVxBQtwbGdfNDzxaeokyC\nRKSt2AATUbkteEu9a8WWNL5676lbmLL2JN7q1rDc27EyNUI9DVwRhKi6kEgkaONTE218auJeYhr+\nL+o2Nh/5B+f+fYJz/56As6053nrdF8FdfGBjbix2XCLRqdUA79u3D4sWLcLff/+N4cOHY9OmTZrK\nRUR6on/bOqhpZ470rJJnNSzJ9wcvoq1v+S4B1dDDHq0a8HJRRVi3q79ajpZY9HZnfBrcHuv2n8bG\nI1dw68EzLNzxF1bu/RtvBNXH+B5+8KhhJXZUItGo1QDb2Njgo48+wvHjx5GZqdrECERERVRtUJvV\ncUJ+wasndNh09ApS0rMBABnZ+WyAX8C6rT8sTI0w+jVfjOzsg4hL97Hu0CVEXk7AxqNXsPnYVfRu\n5Yn3+/nD151fmCP9o1YDHBQUBAA4f/48CykRVRkrs5K/2JObXwC5XIChgRSCAHTxd0dn/1pVnE67\nsW7rH6lUgk5NaqFTk1q4Gp+MdYcuYe+pWOz/8zb2/3kbXZu544N+TdGsjvpfziXSFRoZAywIgiY2\nQ0RUbinp2fg34ZnSss3HrsLX3Q4GUglq2pnDzcFCpHTaj3VbP/m622PFux3x4ZDm+P7gRWwPv45j\n5+Nx7Hw8ujZzx0dDmvOMMOkFjTTAEknZM9DowjUJZbLCqXGZVfN0Ka8uZQV0K68ms/4VG4vFv/4N\nW0tjDA70gZOtBaYMboO2Dd3U3naRorzVUVl1W9ueT9r6PNfVXPb29vhuuge+eKszVu0+g7X7z+PY\n+Xgc/zseQzv64vORHeDtovkvmurq8RILc6lGlZpdZWeA58+fr/h3YGCg4mM4IiJV/BP7GLsirkEq\nlaBZXSc8eZYJP09HNPZy0sj2IyIiEBkZCQAwMDBAYGCgRrarbcqq26zZ+sHJ1hxfjeuEDwa2xOL/\nO4UfD13AjvCr2B11HZP6NcfHI9rC2txE7JhEr1TRml1lZ4AnTpyodFvVaWarQkWnwBWDLmUFdCuv\nLmUFdCtvWVnlcgHZuflKyy7eScLemFg4WhdeQzgjOw8ju/gUmwZaU7+/n58f/Pz8FHmjo6M1sl1t\nU1bd1raara3P8+qSyxDAnKHNMKpTXSzbfR67om5iRegZ/HTsImYPbYFhQfVgIFX/OsLV5XhVFeYq\nW0VrtlrR++7OAAAgAElEQVQNsFwuR25uLvLz81FQUICcnBwYGhrCwMBAnc0SkR5ISE7HvSdpSsvu\nJaUh4uJ9NPRQ/ljt/b7+cOV4Xo1g3abSuDlaYvk7QRj7ekN8/tMpnLnxGB+uj8JPYdewZHwH+NV2\nEDsikUao1QBv3boVY8eOVdzetm0bQkJC8Pnnn6sdjIiqp/wCOX44dBHRVx7gra4NYST776xSTTtz\nfBHcGo7WZiImrN5Yt6k8Gnk6YPdnfbD/z9uYv/00Lt5JQs/P9uLtHo0wY2AzmJlU3/HxpB/UaoDH\njBmDMWPGaCgKEVU3l+OSsS/mFkyMCkuNqakpCuQCTA2l+G5SZ1iYyso1hIo0h3WbyksikaBfG2+8\n1tQdi3edxcYjV/D9bxfx25nbWDw+EIF+rmJHJKowToVMRBrz4mxuT1KzsGLPeXwyvKVivK42jRsj\novIxN5Fh7sg2GNC2Dj5cH4mr8U8xfOEhjH29Iea80RKmxmwlSPfwWUtE5XYhNhEZ2XmvvH/twX/Q\n/oWzQjMGBRT7shoR6SZ/b0ccmj8Aaw5cwPI957Hx6BVEXLqPVe91gr+3o9jxiFTCBpiIyuXJs0ys\n/e0fjOna8JXrzB/dlg0vUTUmM5Ri6oBm6OLvjg/WhuNmwjP0DdmHmYMDMLmPP6RSDmki3cAGmIjK\n5WlaNsyMZThxIR59W3ujkSe/DU6krxp5OuDwlwOwaOdf+PHwZXy98yxirj7Eqokd+SVW0glsgIn0\nVHpWLuSvmAth/5+xiH+SBmOZ8qWx3BwskJNXAH5vjYhMjAwREtwGQY3cMOX7k4i8nICuH+/Gqomd\n+AU50npsgIn01ISVx2FoIFUas/uiaQObwdSIJYKIStepSS0cXTAQk9eEI+baQ4xYdAgzBwXgg35N\nOSSCtBZf3Yj0zL6YWNxPSoOVmTGuxCdj64fdxY5ERDrO2dYcO+b0xIo9f2P5nvNY8us5XLidiJXv\ndoS1ubHY8YiKUXtew/v376Njx44wNzdHQEAArly5oolcRKRhn2z+A8tCz+HcrScY280Py98NwtEF\nA8WORVWMNZsqi4FUihmDArB1ZndYmxnh2Pl49PxsL67feyp2NKJi1G6AJ0yYgMaNG+Pp06cYNmwY\nhg0bpolcRKQBGdl5mLvtTywLPYe2vi6YMSgA80a2gamRoeKH9AtrNlW2zv61cPirAfB1t0Pc4+fo\n/cU+/HbmjtixiJSo1QA/f/4cx44dw+zZs2FsbIypU6fi7t27uHz5sqbyEZEKkp9nIeLifcXPjHWR\naOzpgBmDAtCrpafY8UhkrNlUVTxqWGF/SD8MbFcHWTn5mLDyOJaFnoP8Vd+8JapiajXAt27dgomJ\nCczNzdGhQwfcuXMH3t7euH79uqbyEVE5Pc/Mxbztp/HgaTrMTGQwM5FhUp8mGNCujtjRSEuwZlNV\nMjU2xKr3OuKzEa0glUjwze7zGP7lHqUZI4nEotbnnxkZGbCwsEBaWhquXbuGlJQUWFpaIiMjo9i6\nRVOgajOZTAaAWSuDLuXVpaxz1ofDwqzwCybpmTno38EXA9s30NpvXuvSsQX+y1tdqFKzXVy18zJW\nLmIHeAXmerV5//sBAJwEIrc2hFN0GGo724gX6iXaWpuYSzWq1Gy1GmBzc3Okp6fDzc0NSUlJAIC0\ntDRYWFgUW3f+/PmKfwcGBiIoKEidXRMRgLSsXGTnFQAA5HI5oi/dw4m/49DaxxWjXm8scjrdFBER\ngcjISACAgYEBAgMDRU6kOarUbKLKEnj/CmpM2Yodnw9E24ZuYschHVfRmq1WA1ynTh1kZWUhISEB\nrq6uyM3NRWxsLOrXr19s3YkTJyrdTk5OVmfXlaLonYw2ZnuZLmUFdCuvLmU9cf426tUqnJEtN7fw\nY0WvmtbwdDTRyvy6cGz9/Pzg5+cHoDBvdHS0yIk0R5Wa/SAhQYSEr6atzx3mUk3RJwuJqZnoPms7\nFo/vgCEd6omcSnuPF3OVraI1W60xwFZWVujWrRsWLVqE7OxsLF++HB4eHoogRFR57iemobaTldIy\nuVyAg5UpfN2162Mp0g6s2aQt3nrdF7n5ckz9PgIL/u8MvxxHVU7tayD98MMPCA4Ohp2dHXx8fLBj\nxw5N5CKiUiQkp+PrXWfxQf+m6NKiAYD/3olLOU8xlYI1m7TBl6PboZ6rLT7dcgprDvyD2IfPsOq9\nTjA3qV7j7kl7qd0Au7m54eTJkxqIQkRlyc7Nx5hlR+HmYIE+rb1Qx8UGhgaFH+QU/ZeoNKzZpC1G\nveaL2s7WeHflcfx+9i76z92PzTO6wdWBY9Kp8vEVk0gHZOXm4/jf8Yi4eB9GhlI8eZaJbWHXMGtD\nlNjRiIgqLNDPFfvn9oOnsxWuxj9Fr8/34ty/j8WORXqADTCRDnj6PBvf7D4HJ1tzTB8YoPj5cHBz\nsaMREamljosNDszth3YNXZCYmoUhX/2GX6P+FTsWVXOcB5VIB7jYm2Ngu7oIuxBf7L6jFx4AALKy\nsgAAbg6WGBYk/reqiYjKy9bCBD9/1ANf/BSDLcevYsr3J3H93lN8/EYLGEh5ro40j88qIh2wI+Im\nLscl4fGzTMVPSnp2ieuam/B9LRHpHpmhFAveaoeFb7WDoYEEa3+7iDHLjiI1I0fsaFQN8ZWSSAf4\nezviflI6Lt5JVCwzNTbEl6PbadX1GImI1DXqNV/UcbHBhJXHceLCPfT6fC82TOuK+m52YkejaoQN\nMJEOaFDLDjcTUpCXX4BerTwBgJcLIqJqq62vCw7N749xy4/havxT9P58H5a/E4TerbzEjkbVRIWH\nQNy4cQPdu3eHra0tPD09NZmJiErQt7U3nqRm4fBfcXiUkgnvmjZiRyIdwppNusa9hhX2h/TDgLbe\nyMzJxzurwrDg/84gv0AudjSqBircAMtkMowYMQJLlizRZB4iKsXyd4Iwua8/Tl9/hGWh57Dg/87g\nfuJz3E98jgfJ6UjLzBU7Imkp1mzSRabGhlg9sRNCglvDQCrBmgP/YNiC3/AoJUPsaKTjKjwEwsvL\nC15eXjh+/Lgm8xBRGcxNZPhsRCtci3+KpaFn8f7qIwCA7OwctPapicl9/UVOSNqINZt0lUQiwds9\nGsGvtgMmfXsCf15/hNfn7Mbq9zohqLGb2PFIR/EqEEQ6as+pWxAEwMfdASGjAzF7WEte/oyIqq02\nPjVxZMEAdPBzRfLzbLy5+DAW7fwLefkcEkGq45fgiHTUnDdaAgAWh17A/lM3FdcBfpGRoQHe78cz\nwkRUPTham+HnWd2xat8FfBN6Hqv3XUDkpftY9V4n1HHh9yKo/EptgENCQjBv3rxiy/v374/du3er\ntKOiSzVpM5ms8Fv1zKp5upRXl7ICwDeTugMA8vLyAACCIOBe4nMAwNp95/Dxlj8BAE62Fpj/VpA4\nIf9H145tUV5dUZ1rtrY+d5irYtTN9eX4rujWqj7GLjmIf24nofune7BofGdM6N0UEolE5e1p6/Fi\nLtWoUrMlgiAI6uzs+PHjePvtt3Hnzp1XrhMWFobw8HDF7cDAQAQFiftCXJKiA1fUSGgzXcoK6FZe\nXcoKFM+bnZuPlhM3wtvFVmm9em52aOPrBnMTGbo2F+dSQrpwbCMiIhAZGQkAMDAwQGBgILp06SJy\nKs3R1Zqtrc8d5lKNsYkJACAnu+SJfFSVmpGNad8dw/awKwCALs1q49sPusPTWbWzwdp6vJirbBWt\n2WoNgcjOzkZeXh4EQUBOTg4kEgmMjIxKXHfixIlKt7Xxov26NKGALmUFdCuvLmUFiueVywWsmdQR\nALAz8ibuJ6YDAG7GJ+JmfCKeZ+Wimae1VmTVRn5+fvDz8wNQmDc6OlrkRJqjyzVbW587zKUal//9\nV5O5loxti8CGzpi9MRph5+PQ7J31+HBwAMZ184OhQfm+6qStx4u5ylbRml3hBjguLg5eXoVnkSQS\nCUxNTdGxY0ecOHGiopskIg2QSiXwdS8sThYmRvBx/2/2pLwCOZp6OYoVjUTEmk3VWZ9WXmjToCa+\n+CkGe2NiMe/n09h7KhaLxrZHE9Y8KkGFG+DatWtDLuc3L4m02czBAUq3M7PzsHzPeVy+W/a79oTk\ndOTly9HY0wFv92hUWRGpirBmU3XnYG2KNZM7Y2D7Opi9MRoX7ySh1+d7MTSwHmYPbYEaNmZiRyQt\nwqtAEOkRMxMZPhneqlzrDl94CAPa1UGj2g6VnIqISHO6+Lsj/OvBWLHnb6z//TJ2RNzEwdN38EE/\nf4zr7gdTI7Y+xAaYiF7BwtQI9xLTcC8xDYf++u8LU3JBQP823qjralvKo4mIxGNhaoRPR7TCm50b\nYP720zhy7i4W7vgLG49cwQf9/DG8UwMYywzEjkkiYgNMRCX6ceprJS6/9eAZNh65gpy8fCyboH1X\ncyEiKuLpbI2N019H5OUELPjlDC7FJeGTLafw3cGLmNK/KQZ3qMtGWE+xASYildRxscGCt9rhxIV7\nWBZ6TrH8eWYu3nq9odK6rvYWkBlywkkiElegnys6fNkfv5+Nw9Jfz+H6/RR8tCEKy0LPYWy3hpgy\npB1sLEzEjklViA0wEVVIZ/9a6OxfS3F7X0wszt58rLh96toDjOjUAM3rOokRj4hIiUQiQY8WnugW\nUBsHTt/G6v0XcC3+KRbu+Aur9/+Dt7o3xpB2nvCuyRnl9AEbYCJSW2JqJq7fe6p0zc38AjkePs0Q\nMRURUXFSqQT92nijb2svRF5KwHcH/0H0lQdYvecsVu85iw5+rhj9mg+6NvMo93WESfewASYitWXn\nFiAlPQeO1qYAgEdPM/Dl6HawMit5kgUiIrFJJBIENXZDUGM33EvJww8H/8YvJy4j6nICoi4noIaN\nKYZ0qIehgfVQx4VnhasbtRrgxYsXY/369Xj06BE8PDzw1VdfoW/fvprKRkRVICEpHdl5+WWut+XY\nVVibG7/y/qLmFwDsrExhIJVoJB9pDms2Ucn86zhj7dQemDmwCX6N+hdbj19F7MNUrDnwD9Yc+Act\n6jlhSId66NPai2/sqwm1GmCZTIY9e/agYcOGOHXqFHr06IELFy7A09NTU/mIqAQHTt9GQUHhpAYW\nFoXjbtPT0yq0rd/O3EHPFmX/P9u8nhP6tvau0D5IO7BmE5XOxtwY47v7YVy3hjj77xPsiLiB/X/e\nxl83H+Ovm4/x+dZTeD3AA4M71EVQIzcOkdBhajXA06ZNU/y7bdu28PLywvnz51lMiTQg+koC/rz2\nCJISTqQ+S8/B6K6+AABbm8KP5lKevfrsbGla1HeGq71FhXOS7mDNJiofiUSCFvWc0KKeE+aObIPf\nztzBr9H/4tTVB9j/523s//M2atiYYlC7uhgSWBf13ezK3ihpFY2NAU5JScHNmzfh5+enqU0S6YzF\nu85q/CP/hOR0LHyrfZnXqLS3twcAJHOWT1IBazZR+ZibyDA0sHAscEJSOnb/cQs7o27i9sNUrP3t\nItb+dhH+Xo4Y0akB+rf1hrmJTOzIVA4SQRAETWxo6NChcHR0xJo1a4rdFxYWhvbt22tiN5VKJit8\n0ubl5YmcpGy6lBVQPe/jlAw8S8+uzEj4/sA52FmaFltuYFDYcBYUFJR7W64OVhjbo4nGsqlCl54L\nupQVKMwbHh6OLl26iB1F43StZmvrc4e5VGNsUnit3Zzsyq3vqlL1eAmCgNPXHuCnY5ewK+Ianmfm\nACicge6NTr6Y0LspGnupfwlIbf07anOu8tbsMhvgkJAQzJs3r9jy/v37Y/fu3QCAOXPm4K+//sLh\nw4dhaFj8pHJYWBjCw8MVtwMDAxEUpH0zSGnrH7Qk2p41v0COnSevKm6r2lTu++MmBgU2qJRsRQwN\npBjYofg+tP3YvkyX8upC1oiICERGRgIofN4GBgbqVANcXWu2tj53mEs11aUBflFWTh52R93A+kMX\nEHP1vmJ5R38PTB3YEq8394K0gp8QauvfUZtyVbRmq30GePny5di+fTtOnjwJc3PzEtcJCwuDj4+P\nOrupEoqPkpOTRU5StsrMevtRKnZF3lRrcH9+gRwSiQRDOtQFANja2gIo/Ni1PCQSCWo7WVV4/+rQ\npecBoFt5dSkrUJg3OjpapxrgsuhqzdbW5w5zqcbF1RUA8CAhQeQkyjR1vG7cf4ptYdexI/ImMrIL\nm8O6LjZ4r3cTDGpfR+XXVW39O2pzrvLWbLXGAG/ZsgU//PADoqKiXllISXyf/xQDaxUu25KamYth\ngfXQ0MNeYxns7Qsb4GRjuca2SUSqYc0mqlz13ewwf3RbzBwcgO3h17HhyBX8++AZpq+LwLcHLmDG\nwGbo29q7wmeESXPUaoDnzp2Lhw8fwsvLS7Hsk08+wezZs9UOpo+ycvIRn/i8xPuW/noODWr99y1T\nU9PCsatZWVllbrd53Rq8fBURsWYTVRFrc2O817sJxndvhL0xt7Biz9+4/TAVk9aEY9W+C/h0eCul\nqeSp6qnVAN++fVtTOfRKxMX7eJKaWWz53cdpyMsvgJ+nQ7H7JvRsjBb1/htQr60fPxCR9mLNJqpa\nMkMphnSoh/5t6uDX6JtYvvtv3LifgpFLfke3AA/MHdkGtRwtxY6plzgVsob9FHYNT54Vb25fFJ+Y\nhukDmxVb3rK+M2pYm8HUmH8WIiKi6kJmKMXwjg0wsF1dbDp6Bd/sPo8j5+4i4uJ9TO7rj4l9mpR5\nyUvSLHZaFXTj/lNsPX4NdpYmSsslAGYMChAnFBEREWktY5kB3u3VGP3bemP+z6exNyYWS0PP4be/\n7mD1e53g484JNaoKG2AAD5LTkZqRC5vnhZfoevYsVen+G/efIuzCPaWrEsgFAW+93hB1XGyqNCsR\nERHpNmdbc6yZ3Blvdm6AD9dH4Vr8U/T8bA9mDW2BCT0a8UtyVUAvG+Ds3Hzsi4lV3D5y7i4Gta8L\ny4zCBjgtLV1pfZmhAT4b0Qo1bDjVFhEREWlGW18XHF0wEHN//hM/n7iO+dtP4/jf8VgzqTOcbNlz\nVCa9bID3xcTiaVo2+rQq/CZ0Bz9XuNhb8ItlREREVKXMTWRYPK4DujZ1x4froxBz7SF6frYH66a8\nhtftNXc5UlKmNw3wp1v+gK1F4Xjd55m5mDuyjciJiIiIiAp1beaBYwsd8c7KMJy+8QiDvzyIFZPy\nMLZHE7GjVUvVsgG+fu8pDp25g9TMXFj9bwKINj4u6NXSU+RkRERERCVztDbDjjm9MPfnGGw6ehUT\nVx7GP7GPMWdoU7VmZ6XiKnw0ly9fDi8vL1hZWcHDwwMLFizQZC6VnbnxCL+cvI5fTl7H8j3n4eth\nj9lDW2DGoADMGBTA5peI9J621W0iKk5mKMWXo9vhmwlBMJYZ4IeD5/HOquPIys0XO1q1UuEGuHfv\n3jh//jyeP3+OqKgofPvttzh27Jgms5Xb5mNXsf73ywhs5IbARm5YNLY9ujevzevpEhG9QJvqNhGV\nblhQPRxeNBw2Fsb4/exdBH99GKkZOWLHqjYq3ADXrVsXNjaFlwDLySn8g1haVv1sJt//dhEp6dlY\nN+U1uNpbwNXeQjHWl4iI/qMtdZuIyqdtQzecWBoMZ1tz/Hn9EQZ9eRCPU0qfbIvKR60BJdu3b4eF\nhQUaNGiAjz/+GK1bt9ZUrlfKys3HpTtJeP+7cCwLPQcAmDag+KxqRERUnBh1m4gqzre2I/aH9IV3\nTWtci3+KgfMP4OHTDLFj6TyJIAiCuhuJiorC4MGDcfToUTRpUvzbimFhYWjfvn2Ft3/5zhP8ee0B\nAOBBchpycvPRpZknOjetXeFtlkQmkwEA8vLyNLrdyqBLWQHdyqtLWQHdyqtLWYHCvOHh4ejSpYvY\nUTSutLqtbs2uDNr63GEu1RibFH5Cm5OdLXISZdp6vF7MlZSaiT6f7MDftx6jrqsdji4egZr2FqLn\n0iaq1OxSB8mGhIRg3rx5xZb3798fu3fvVtzu0KEDBg4ciG3btpXYAAPA/PnzFf8ODAxEUFBQmeGK\nnL7+ABH/3MX0wa3Qs5U3bC1MYGosK/fjiYjKKyIiApGRkQAAAwMDBAYGipxINZqq2+rUbCLSPAdr\nM/y28A30mP0L/ol9gu6zf8HRxSPgZGsudjRRVbRma+QMMAC88847sLKywpIlS4rdFxYWBh8fH5W3\nKQgCvvgpBtbmxniWnoP3ejeGSyW+29GliTB0KSugW3l1KSugW3l1KStQmDc6OrpangEGXl23K1qz\nK5O2PneYSzUurq4AgAcJCSInUaatx6ukXE/TsjH0q99w7d5T1HO1wa5PesPB2lT0XNpAlZpd4THA\nq1atQkJCAgRBQExMDHbs2IHu3btXdHMl+mh9FNxrWGHGoADMH922UptfIqLqrirqNhFVLjtLE+yY\n0xP13WxxM+EZRi75HRnZ2jUUQRdUuAG+ePEiWrVqBUtLS4wePRpLlizR6FmSo+fvQgCQl1+gsW0S\nEemzyq7bRFQ17K1MsWNOT9R2ssLFO0l4e8Ux5LJfUkmFL5S7fv16TeZQ2P9nLP5NeIbHKZn4aEhz\nWJhwrC8RkSZUVt0moqrnaG2Gn2f1QL+Q/Yi4lICZP0Zi5bsdIZFIxI6mE7RmXj1BEFAglyMjOw+p\nGTno3coTNWzMYMYGmIiIiKiY2k5W2PphN5gZGyI0+hYW7vhL7Eg6Q2sa4FNXH2Lmj1F4lJIJW0sT\nOOv5txqJiIiIytLEyxE/Tn0NhgYSrDnwD7Ycvyp2JJ2gFXMFP03LxrG/76JfGy90bFxL7DhERERE\nOqNj41pYMj4Q036IwGdbTsG7pjXaN3QVO5ZW04ozwE/TspH8PBttfFzEjkJERESkc4YG1sPkPk1Q\nIBfwzsow3H6UKnYkrSZ6A5ybX4A1B/6Bv5cjjGUGYschIiIi0kmzhrbA68088CwjB2OWHkFqRo7Y\nkbSW6A3wgLkH4GpvgaGB9cSOQkRERKSzpFIJVk/sCJ9adoh9mIr3Vochv0AudiytJHoD7Otuh5MX\n7+PKXe2aTYSIiIhI11iYGmHTjNdhb2WCiEsJ+HonrwxRErUb4JSUFDg6OmLkyJEqP/bo+buwMDXC\n1g+7IaCuk7pRiIioDOrUbCLSDbUcLbHug9dgIJXgu4MX8duZO2JH0jpqN8Bz5syBl5eXyhdezi+Q\n4/9O3sCcN1rCztIEMkPRT0YDAK5duyZ2hHLTpayAbuXVpayAbuXVpazVUUVrtjbQ1ucOc1UP2nq8\nKpqrtU9NfDaiFQBg2g8RuHk/RZOxtPZ4lZdaXee5c+cQFxeHnj17QhAElR77y8kbaOTpoDWNbxFd\n+oPqUlZAt/LqUlZAt/LqUtbqRp2arQ209bnDXNWDth4vdXKN7+6Hfm28kZGdh3ErjiEtM1crcmmD\nCnefgiBgypQpWLZsWYUK6dX4ZNRxsano7omISAXq1mwi0j0SiQRLx3dAAzdb3H6Yimk/RPD///+p\n8EQYGzZsQOPGjeHr61uuj9Ls7e2Vbtews8aIrs1gpEWXPpPJZOjcuTNsbLS/MdelrIBu5dWlrIBu\n5dWlrEBh3upC3ZotNm197jBXxfD5VT6ayGUPIHTeULR9fzMOn43DtohYTB3USvRclUGVml1qAxwS\nEoJ58+YVW96xY0fEx8cjJiYGAMr1biI6Olrpdvd6hjhzOqbcQYmIqHSVWbOJNOL48cL/8vlV5UKn\ntfjfv/L4/zcAiVCBc+H//PMPmjZtWmy5v78/zp8/r5FgRESkGazZRETKKtQAv2zu3LmIjY3F1q1b\nNZGJiIgqEWs2Eek77boEAxERERFRJdPIGWAiIiIiIl3BM8BEREREpFfYABMRERGRXqnwdYCJiKh6\nS09Px5QpU+Dv74/3339f7Dg4ePAgfv/9d6SlpcHc3ByvvfYaBg4cKHYs7Nu3DydOnMCzZ8/g4OCA\n4cOHo3nz5mLHwoMHD7Bp0ybcunULZmZmWLNmjah5kpOTsXr1asTGxsLFxQWTJ09GrVq1RM30119/\nYe/evYiLi0O7du0wceJEUfMUKSgowNq1a3Hp0iXk5OTA09MT48aNg5ubm9jRsGrVKly+fBk5OTmo\nUaMGhg0bphXP9yLXrl1DSEgI3nnnHXTu3PmV6/EMMBERleiXX36Bk5NTuSbOqAoBAQH4+uuvsWXL\nFsybNw9HjhzBxYsXxY4FAwMDzJw5E1u2bMGECROwevVqPHnyROxYMDAwQPv27REcHCx2FADAunXr\n4O7ujo0bN6Jt27ZYsWKF2JFgbm6Ofv36oVOnTmJHUSKXy+Hs7IyFCxdi8+bNaN68OZYsWSJ2LABA\nv379sGbNGmzZsgUjR47EN998g5ycHLFjASh847B9+3a4urqWuS4bYCIiKub27dtITExE06ZNtWbq\n1Jo1a8Lc3BwAkJeXBwAwMTERMxIAoHfv3oozmfXr14eTkxNu374tcirAyckJQUFBcHR0FDsKMjMz\ncfHiRfTv3x8ymQy9evVCYmIi4uPjRc3l6+uLli1bwsLCQtQcL5PJZBg8eDDs7OwAFE5m8+jRI6Sl\npYmcDPDw8IBMJoMgCMjPz4eJiYnWvEk+fPgwmjVrBmtr6zLXZQNMRERKBEHApk2bMGrUKK1pfotE\nR0dj5MiRmDp1Kvr374969eqJHUlJeno6Hj58CHd3d7GjaJVHjx5BJpPBxMQEn3/+OZ48eQInJyc8\nePBA7Gg64ebNm7Czs4OlpaXYUQAA69evR3BwMFavXo1Zs2bByMhI7Eh49uwZIiIi0Lt373KtzwaY\niIiUnDhxAh4eHnBzc9OaMztF2rdvj59++gkhISHYvXs34uLixI6kZN26dQgKCoKLi4vYUbRKTk4O\nTExMkJWVhYSEBKSnp8PU1BTZ2dliR9N6mZmZ2Lx5M0aNGiV2FIXx48dj69atGDZsGFavXo3c3Fyx\nI2Hr1q0YMGAAZDJZudbnl+CIiPTQzp07ERoaWmy5r68vkpKS8NVXXwFAlZ8BflWuFi1aYObMmYrb\nPvxkJuAAACAASURBVD4+aNmyJaKiolC7dm2tyLV9+3ZkZGRgypQplZ5HlVzawNjYGNnZ2bC3t8eG\nDRsAAFlZWVoxhEWb5eXlYcmSJWjXrh3atGkjdhwlBgYG6N69O44cOYLLly+jWbNmomW5fv06EhMT\n0bZt23I/hg0wEZEeGjp0KIYOHVpseVxcHGbNmoW3335bafn9+/fx9ddfi5arJFXZnJeV6+DBg7h0\n6RK++OILGBgYaE0ubeHs7Izc3Fw8ffoUdnZ2yM/Px+PHj3mmvBRyuRwrV65EzZo1tfpvrA3DpG7f\nvo2bN29i2LBhimVXr17FvXv3MHr06BIfwwaYiIgUateujR07dihu79q1C48fP8bkyZNFTFXo0KFD\naN26NWxtbfHvv//i1KlTmDFjhtixcPLkSRw/fhzz5s3TujOaubm5KCgoAFB4NlEikcDQsOpf+s3M\nzNCkSRPs3bsXwcHBOHToEBwdHUUfKy2Xy5Gfnw+5XA65XI68vDwYGBhAKhV/hOi6desgkUgwfvx4\nsaMoPHv2DOfPn0fr1q1hbGyMEydOIDU1VfSx+D179kTPnj0Vt+fOnYsOHTqUehk0NsBERKQT4uPj\nsX//fmRkZMDOzg7BwcFo1KiR2LHw66+/IiUlRelNwsCBA9G/f38RUwFPnjxRun5zcHAwfH198cUX\nX4iSp+gScWPHjoWrqyumTp0qSo4XRUZGYu3atYrbUVFRGDJkCAYPHixiKiAxMRHh4eEwMjLCmDFj\nFMvnzJmDBg0aiJZLKpUiOjoaP//8M/Lz8+Hm5oZZs2Zp3VU0ykMiaMO5ayIiIiKiKiL+OX4iIiIi\noirEBpiIiIiI9AobYCIiIiLSK2yAiYiIiEivsAEmIiIiIr3CBpiIiIiI9AobYCIiIiLSK2yAiYiI\niEivsAEmIiIiIr3CBpiIiIiI9AobYCIiIiLSK2yAiYiIiEivsAEmIiIiIr3CBpiIiIiI9AobYCIi\nIiLSK2yAiYiIiEivsAEmIiIiIr3CBpiIiIiI9AobYCIiIiLSK2yAiYiIiEivsAEmIiIiIr3CBpiI\niIiI9AobYCIiIiLSK2yAiYiIiEivsAEmIiIiIr3CBpiIiIiI9AobYCIiIiLSK2yAiYiIiEivsAEm\nIiIiIr3CBpiIiIiI9AobYCIiIiLSK2yAiYiIqESbN2+GVCpFfHy82FGINIoNMBEREb2SRCIRO8Ir\nrVixAvv27RM7BukgiSAIgtghiIiISPvI5XLk5+fDyMhI7Cglql27Njp37oyNGzeKHYV0DM8AExER\nUYmkUqnWNr9FeB6PKoINMBERESnx8/ODVCpV/JQ0BlgqlWLu3LlYunQpatWqBRsbGwwYMADJyclK\n640ZMwaenp44dOgQfH19YWpqCn9/fxw+fFhpvZMnT0IqlSIyMrLExxcpGpdclGvLli1KWV9+PFFJ\nDMUOQERERNrl66+/RmpqKiIjI7Fu3bpXrrd9+3bY2tpi9uzZiIuLw4oVKzBhwgSEhoYq1pFIJEhO\nTsabb76J9957D87Ozvjxxx/Rr18/REREoE2bNmXmeXEcclBQELZt2wZBEDBt2jT4+vpiwoQJivsb\nNGhQwd+a9AkbYCIiIlLSq1cvAEBubm6pDXBGRgYuXbqkGCaRkpKCbdu2QS6XQyot/JBZEASkp6fj\nxx9/xLhx4wAAb775Jjw8PDB37lz8/vvvZeZ5cZiDp6en4ozwp59+Ci8vL4wYMaJivyjpLQ6BICIi\nogrp1auX0hjhpk2bIjc3F0+ePFFaTyqVKjWp9vb26NatGyIjIyGXy6ssL1ERNsBERERUITVr1lS6\nbW5uDqDwzPGL7O3tYWpqqrSsVq1ayM7ORlJSUuWGJPr/9u48MKry3v/4Z2YyWUlCMiQhIYR9CYQd\nZHUiImrRCqLC1YJSte0t1aqVlvuzvXXB5Yd2UdHaYqugV6wboqUoKktC3NiUgIAECEsIhDBJyL7P\n/QPJNQYhgSTPSeb9+iszPZO8c4Llm4dnzjkDBmAAAHBeTm9zOJezXakhICDgrK+tqalpUhPQGAzA\nAACgRXk8HpWVldV77tChQwoLC1N4eLgk1W2lKC0trXdcdnb2996Mw8o36YC1MQADAIAW5fV69cor\nr9Q9PnHihFavXq2JEyfWPRcfHy9J2rhxY91zhw8fVlpa2vd+3tDQUGVnZ7dAMdo7rgIBAADqpKen\nKz09XZL06aefSpLefvttuVwuSdLll1+u6OjoJn3ODh066Ne//rX279+vmJgYPf/886qqqtJ9991X\nd0xCQoIGDx6shQsXqqKiQh06dNALL7yg3r17N1gVPm38+PF6/vnn9eijj2rIkCFyOBwaPXq0IiIi\nzudbhw9hAAYAAHXefvttPfjgg3WPbTab7rnnnrqP161bd9YB+EzbElwul5599lnNmzdP+/fvV//+\n/bV8+XKNGjWq3nHLly/Xbbfdpqefflrx8fF69NFHtWrVKqWkpJzxaz388MPyeDz6wx/+oIKCgro+\nt9t9Pt86fIjNyz0EAQBAC5kzZ45SUlKUmZlpOgWowx5gAADQonizGqyGARgAALQo/rEZVsMADAAA\nWozNZmMFGJbDHmAAAAD4FK4CAQCoZ82aNaYTAOC8TZo06ZzHMAADABpITEw0nVCPy+XS8uXLlZyc\nbDqlHrqahq6moatpXC7XWW+c8m3sAQYAAIBPYQAGAACAT2EABgC0CVbblnEaXU1DV9PQ1TIYgAEA\nbYJV/8Klq/GqqmsU3aW7yiqqNe/5VN342CqVlleZzpJkzfMl0dVSeBMcAABoVp7CMhWVVSmmY7C2\nHzih8BB/9YuPVNqOw1r5aYb87V7FRoaoV2x4g9dWVNWovLJaQQF+8vdzGKiHL2AABgAAzeqv/05X\nWWW1po7ppeUf79Wx/BKN7tdZ44f01LQJ/fSvtJ2KjQxRVXWtnlzxha4d11uJCZGSpLfSMrTrUJ6S\nunfSzOS+hr8TtFcMwAAAoFkF+vvp0qEJWrkxUxMGxmny8G5659N9en/jPt1y+WDNnzGq7tgdB04o\nIztfveLC61Z8fzimpzKPFZrKhw9gAAYAABfkw60HtW5blorLK9UtOky7D+fp3utGaGxibN0xM9x9\n5XK5JEkej6fu+ZiIYKVsz9J/Pr1G3WPC1DuuY6v3w/cwAAMAgPO2YNnnctht+s2MkQoNcsphb9r7\n66PCg/WLHw6VJP3xrS0tkQg0wAAMAADOy5f7clVeWa1H5ow3nQI0CZdBAwAAjfbxV9la+PomfZ2V\np6Uf7dTUsb2a7XPHRoYo40iBgvxPrc8teudLVoXRIlgBBgAAjbZtf66mj++t11L26JLB8bqoX+dm\n+9w3TewvSco9WarXUr9Wr9iOyiuqabbPD5zGAAwAAJokvlOofnfT6Bb7/FHhwXr4llPbKlgBRktg\nCwQAAAB8CgMwAABolJz8UuUXl7f6162oqlFeUbnKKqtb/WujfWILBAAAOKeHl32uzJyTunJkdzn9\nWm/9LLpjsH635GN17BCgvdknddmwBP3o0v6t9vXRPjEAAwAaOH3DAqtwOp2S6GqsluiK7BiqP985\n5YI+x/l03T1jQr3HC17e0Ozn25d+js3B6l2NwQAMAGhgwYIFdR+73W4lJycbrIFJhSUV2pOVJ6/p\nkG9k5Rbp5Q+3a/bkQaZTYAEpKSlKTU2VJDkcDrnd7ka9zub1eq3yZxoAYAFr1qxRYmKi6Yx6znQL\nXSvwha4NO44oPTNXV47srl6xF3ab4uboyskv1V9WbtODs8deUMu3+cLPsTlZuSstLU2TJk0657G8\nCQ4AAJzVyD4xFzz8NpeYiGAN7tFJ9734sdIzc03noI1iAAYAAGeUe7JUWSeKTGc0cN2EPrrqoh7K\nPFaogpIK0zlog9gDDAAA6pRVVuuZd7+U3WbTjgMeXTu+l3rHWWP199t6x3VU1oli3f/yp+ri6qDf\n3DDSdBLaEAZgAABQp7KqRh1DAvSTH1j7TWYxEcGamdxXM5P7crc4NBlbIAAAAOBTGIABAADgUxiA\nAQCAJOn9zQf0+BubNbCbtW5wcC6ewnK9uSFDVdW1plPQRjAAAwAASVJVTa1uvixR4wbEmU5pkl9O\nG6o9R/L15Iqt2ptdYDoHbQADMAAAaNM6R4Ro/oyRmjysGwMwGoUBGAAAqKq6tk1vIXDY7fJzMNag\ncfiTAgAA9NhrG5XtKVansCDTKUCL4zrAAABAIYFO3XHNUNMZzaa8slq1tV4F+vvJbreZzoHFsAIM\nAADajZMllSouq9Qjr27U3X9L0dH8EtNJsCBWgAEA8FEl5VVa8+Uh+fs5lHWi2HTOBYtzhWhzRo5+\n+dx6JUSHauKQeNNJsCgGYAAAfNSJwjIdyCnUpUMSdNsVA03nXLDI0EDNmTxAcyYPkCS9un634SJY\nFQMwAAA+aH36Ya3dlqWLB8YpqXvbuvEFcKEYgAEADbhc1hqInE6nJLoaqzFd+WWH9dvZyYpzhbZW\nVqufr/CwML28LkMzLhmgUf2+/+YebfnnaILVuxqDARgA0MCCBQvqPna73UpOTjZYA5yf2ZMH6dDx\nBK394sBZB2C0XSkpKUpNTZUkORwOud3uRr2OARgA0MDcuXPrPfZ4PIZKTjm90mS647vacldJSbHy\n8/MVoMrWyjJyvgryi1VSXHLWr9mWf44mWKkrKSlJSUlJkk51paWlNep1XAYNAAAfUlJepUde/Vx7\nswsU6M86GHwTf/IBAPAhc59Zqymjumtmcj/TKa3mg60HFRbirymjephOgUWwAgwAgA8Z3KOTTw2/\nca4Q/f3uydp5MM90CiyEFWAAAHxAVXWtCksrVOv1mk5pVTabTTabdPB4oRa+vkl+DrtmT0pUdMdg\n02kwiAEYAAAfsOuwR6+s3a3p43ubTjHidzeOVq3Xqy/3HVfuyTIGYB/HFggAAHzEpGEJGt0/1nSG\nETERwYqNDDGdAYtgAAYAoJ1b++Vh/XP9HgU6HaZTAEtgCwQAAO3crsMe/fePRiuIy54pNrKD3krL\n0PbME/qPS3znzYCoj/8SAACAzxjaK0q948L1P2t3m06BQWyBAAAAPmfX4Txt+vqY6QwYwgAMAAB8\nSlCAn342ZZBWbzloOgWGMAADANCObdhxRDsOeExnWIrDbteABJeCAtgJ6qsYgAEAaGfyi8q1Yfsh\nZeUWKXV7luZdP0IBflwB4ruOF5TqvU2ZpjNgAAMwAADtzNaMY9q2L0dPvLVFTj+HesV2lN1uM51l\nOT+7arBSth8xnQEDWPsHAKAdGta7s268uKfpDEvr2Tlcw3pF61d/+VCzJw9SQqTTdBJaCQMwAADw\nWTOT+2pHVrEKSyskBmCfwQAMAGjA5XKZTqjH6Tw1mNDVOH5HSuXn52e5Lsuer2PlkqzXZdXzZfWu\nxmAABgA0sGDBgrqP3W63kpOTDdagKXYdPKFPv8rSxGHdTacALS4lJUWpqamSJIfDIbfb3ajX2bxe\nr7clwwAAbcuaNWuUmJhoOqOe0ytNHo+1Ludlxa4n3tysmZcOUZ/4SFWUFpnOqceK50uSjhbW6OUP\ntys6zF9zJg8wnVPHqufLyl1paWmaNGnSOY/lKhAAALQjdptNw/p0Vocgf9MpbUZSj2gt/OkkeQrL\nTKeglTAAAwAAwKcwAAMA0E58sjNbmcdOms5os3LySzXv+VSt/Hy/qqprTeegBTEAAwDQTqzeclD/\nedUQ0xlt1uO3X6y7pg3T1r3HVVFVbToHLYirQAAA0MaVllfp/7++SUH+fkrqbq1LU7U1XaNCFd0x\n2HQGWhgDMAAAbVxlTa0SosN0+5VJplOANoEtEAAAtGE5+aVKSc8yndHuFJZWsg+4HWMABgCgDfv8\n66OSpGvG9DRc0n6MGxCrv7+/Qx99cdB0CloIAzAAAG3U57uPKm1HthK7RrJvtRkN7hGl6yb0MZ2B\nFsQeYAAA2qiU7Ud0z/Th6hQWZDoFaFNYAQYAoI1y2G2KjQyR04+/zptbVHiQdh7K04Jln5tOQQvg\nvxgAANqQvKJyPf7GZs3542oVlVWazmm3ojsG697rRsjPYde9i1NM56CZsQUCAIA2pLC0Uj07h+s3\nN4w0neIT/t/MUfrjW1tMZ6CZsQIMAAAAn8IADABAG7HzkEevrv/adAbQ5jEAAwDQRmzJOK4Z7j66\nZizX/G1NrtBA3f3X9crJLzWdgmbCHmAAQAMul8t0Qj1Op1MSXSEhIeoaG6NYV4ezHsf5appzdd17\no1tLVm9Th7AwuVzhlukyxepdjcEADABoYMGCBXUfu91uJScnG6wBrOGfa3dq6vi+6tfVWoOfL0tJ\nSVFqaqokyeFwyO12N+p1DMAAgAbmzp1b77HH4zFUcsrplSbTHd/Vml3vfrZPKzZkaHy/TvJXhWW6\nmqItd102qLP2HMnXhxt3q1Nwf8t0mWClrqSkJCUlJUk61ZWWltao1zEAAwDQBmQcKdBLv77SdIbP\nCgrwU6dw7rjXXvAmOAAALO7fGzN1xFNsOgOSPthySP/emGk6AxeIARgAAIs6nFukR/+5UZv35Oiu\nacNM5/i8uMgQvfCryfrqoEd3/3W96RxcAAZgAAAsKq+oXKP6xuj+WWPULTrMdI7Ps9lsstls+s0N\nIxUVHqRl63Yrr6jcdBbOAwMwAABAE/3imqGSpGxPieESnA8GYAAAgCbqGBKgyNBA0xk4TwzAAAAA\n56Fn53D967N9+tuqdNMpaCIGYAAAgPPQNz5Cv5w2TPuOntSOA+aviYvGYwAGAMBivjro0fPvbdf/\nrNklu91mOgdnEejv0PRxvfVm2h7TKWgCboQBAIDFvJa6R5cPT9CtVwyUw85alZU57HaNSYzVxzuz\nTaegCRiAAQCwmPBgf00Y2MV0BtBu8WslAAAW4fV69dmuozpZUmE6BU10sqRCmzNyTGegkVgBBgDA\nAr7Yd1xvf7xXEaGBuv7iPqZz0ERTx/bSkg93amSfGNMpaAQGYAAALKCotFI/HN1To/p1Np2C8zCi\nT4zWbcuS1+uVdOqucbAutkAAAGDY66l79P7mgwoKcJpOwQUYmxirXy1O1WOvbdL2zBOmc3AWrAAD\nAGDY4dwiPfrj8aYzcIHGD4zT8D7Ryi0o1eupGRrUo5PpJHwPVoABAACaSZC/nxKiw8QOCGtjBRgA\n0IDL5TKdUI/TeWprQHvsWr5htzKOFjXr99aez1dLaImuoKCgC/58vnS+msPprsZgAAYANLBgwYK6\nj91ut5KTkw3WtE/FZZV69JWPVVxeqUV3XmE6B83saF6xfvuPdXrktommU9q1lJQUpaamSpIcDofc\nbnejXmfznn67IgAAktasWaPExETTGfWcXmnyeDyGS+q7kK68onKt+GSvbr0iqbmz2uX5akkt1fX4\nG5v186sGq0OQ87yuCuFr5+tCuVwupaWladKkSec8lj3AAAAALSCpu0t3PrdOxWVVplPwHWyBAACg\nlaV9dUQfbD2kMf255m97NmVUDx3OLTKdgTNgAAYAoBU9vOxz+Tsd+uU1Q9UpPMh0DlpYz87hevyN\nzZo8PEHuQfGmc/ANBmAAAFpRUICf7r1uhOkMtJLJw7spvlOoFr37pUorqnXlyO6mkyD2AAMA0GoO\n5BTqZGml6Qy0ssSESP3ljkv11UFrvWnMlzEAAwDQSv789laNT4w1nQFD9hzJ1zPvfmk6A2IABgCg\nxeUXl+vPy7cqPCRAV/BP4D7rb7+8TCcKy/RWWoZKyrkyhEkMwAAAtLCC4grFR3XQAz8aYzoFhs29\neohy8ku18I3N2rY/13SOz+JNcAAAtAK7zSa7vek3Q0D7Et0xWHN/OETZnmKtT8/SkJ5RppN8EgMw\nAAAt5KMvDmnVpkyFBfvr6tE9TefAYrI9JcrKLVJ8VKjpFJ/DAAwAQAvZkpGj/75ptMKDA1j9RT0R\noYEa2C1Sv136iYb2itI91w43neRT2AMMAEALmPvMWgX6+ymiQyDDLxoI8vfTD0b10NJ5V6i21ms6\nx+cwAAMA0IxKy6uUnpmr7jFhumvaMNM5aAOqa2qVnpmrsopq0yk+gwEYAIBmtOdIgVZvOahrxrDn\nF41zzZheWrXpgPYdLTCd4jPYAwwAQDP526p05RWWa/qE3uoXH2k6B21EYkKkDh0vNJ3hU1gBBgCg\nGby3KVOf7Dyqe68fwfCLJgsOdOqfKV/rg60HTaf4BFaAAQANuFwu0wn1OJ1OSdbsqq316vG3vlRx\nWZWW/Nc0ucKCZLOZfdOblc+XRNeZTEt2acq4gXrklY+199hObduXo3ceudF415lY4XydyemuxmAA\nBgA0sGDBgrqP3W63kpOTDdZY066DJ3TweJHGDoxXWEiAFv50kukktHH+TocenOOWJD308gZl5Raq\nU3iwHFxE5HulpKQoNTVVkuRwOOR2uxv1OgZgAEADc+fOrffY4/EYKjnl9EqT6Y5vW/LeFnWP66QA\np01lpWWWarPi+ZLoaoqLE6P01JufafLIHhrc1Vo3yrDS+UpKSlJSUpKkU11paWmNeh0DMAAATfTb\nJR/Lz2HXFaN6afmG3ZowMM50EtqZwT2ilBAXo5c+2K4PN5Yr92SZfn/TaAUF+BnfYtMe8CY4AAAa\nqayyWks/2imvV3pw9lj1jO2oeTPGaFS/zqbT0A71iovQg3Pcuve6Ebo4qYt+/sxaFZVVmc5qFxiA\nAQBohK17j+vx1zerttar+TNGms6Bj7nqoh6aMDBOz7z7pb7OyjOd0+axBQIAgEY4nFukH13aX73j\nOppOgY/6yQ8GadehPO3NPsml9i4QAzAAAGfxzqf7tD49S6HB/rqIrQ4wLDI0UO9+tk/p+3N1g7uv\n8orKNbhHJwX6M9I1BWcLAIAzOHGyTIvf267yqhrNnzFSnSNCTCcBiokI1vwZo/Tnt7fqpY92yunn\nUGxkiLpGWetKEVbHAAwAwHfkF5drW2au+neN1PTxvU3nAA3cc+1wSdLKz/fr+fd36MoR3TRuAFcj\naSwGYAAAvuXPy7cq92SZhvWO0tj+saZzgLO6enRPuQfF6y8rt2nVpkz99j9OXSoNZ8cZAgD4tNLy\nKh0+UaT3Nh3Qxq+PaeKQrrpn+nDTWUCjhQX7679mjNKydbt1+5Mf6seXD9RF/TorLNjfdJplMQAD\nAHyO1+tVSXmVnlrxhbJOFGtknxhdO7637pw6VA47VwhF23TTxP5yJ3XRlr3H9dSKL1RUVqmrRvVQ\n8uB402mWwwAMAPAZ5ZXVyjhSoH9vytTuw3m6dlxv/fbG0aazgGYTHxWq+KhQ/XB0Tx3xFOtvq9K1\nalOmFtwyTk6HnbvIfYMBGADQrtXWelVQUiGH3aanVnyhmIhg/XjyQMVEBJtOA1qM3W5T16hQPXzL\neG38+pj++u90VVXXqryyWpcM7qriskpdOjRBTj/f/BcPBmAAQLuz85BHKelZ2nU4TwF+DtntNvWO\n66g4VwfdfmWS6TygVV3Ur3PdNaxfWL1DOw95dDSvRJ9/fUzDe0draM8oxfvYZdQYgAEAbcKuXbsU\nHR39vf97tqdYmzNy9OmuowpwOnTpkK76+dVDjHeZQlfT+ErXrVec+gWwttarwrJKbdx9TM/8a5uO\n5pWoR+cwTR3bS6FB/goK8FMXV4dW62ptDMAAgDbhq507tXq7R/5+dnUI8le2p1g1tV6t3XZYUWFB\nCg8J0OThCbp3+gh1Cg9qtS6rDgJ0NY2vddntNnUMCdDlI7rp8hHdJEkHcgq1JSNHfg67/vXZfpVV\nVmt472hV19Rq39GT+sHI7rr2m+tiW/V8NRYDMADAiMLSSuWeLFVpebU6Rwbr3U/3q6KqRp6iclVU\nVctutyvz6Ek5HDbFdeqorMPHddOV3SVJRWWVum5CH/n72XXrFQPl7+cw+80A7UD3mDB1jwmTJE0d\n26vuea/Xq6qaWi1bu1u/eGatoiLDtHdfjl7a+oHsNptiI0NUU+tVRIdA9YwNl7+fXX4Ou/rGRyg4\nwE8BTociOgSa+rbOyOb1er2mIwAA1rFmzRqtyzz3Xw2nj9ifXaCwEH/ZbTZVVteotLxa1TU16p/Q\n6f+O9Xq1bV+OBvWMlv2bd6Hn5Jeod5cIhYcE6tDxkxrRJ1YTh3VTVXWtQgKdKquolp/DruBAp5xO\np3Jzc9WxY8eW+JbPG11NQ1fTtIWu/G9+YY3oEKhj+SWqqq5RRVWNDuacVFZuoex2u7ZmHFVsZMPt\nFIeOF6prdJjOdl2K08ecZtP//X/Pt1/nleT0c+jirrWaNGnSOb8HBmAAQD1r1qwxnQAA540BGAAA\nAPgO37z4GwAAAHwWAzAAAAB8CgMwAAAAfAoDMAAAAHwKAzAAAAB8CjfCAACcUXFxse666y4NHTpU\nd955p+kcrVy5Uu+//76KiooUEhKiyy67TNOnTzedpXfeeUdr165VQUGBOnXqpBtvvFEjR440naXs\n7Gy9+OKL2rt3r4KDg/Xss88a7fF4PFq0aJH27dunuLg43XHHHeratavRpk2bNmnFihU6cOCAxo8f\nr7lz5xrtOa2mpkbPPfectm/froqKCvXo0UO33Xab4uPjTafp6aef1o4dO1RRUaHo6GjNnDnTEn/e\nT9u1a5ceeOAB/exnP9Oll176vcexAgwAOKNXX31VMTExstnOdpn61jNixAgtXLhQS5cu1UMPPaTV\nq1crPT3ddJYcDofmzZunpUuX6qc//akWLVqk48ePm86Sw+HQhAkTNGvWLNMpkqTFixcrISFBL7zw\ngsaNG6cnn3zSdJJCQkI0depUTZw40XRKPbW1tercubMee+wxLVmyRCNHjtQTTzxhOkuSNHXqVD37\n7LNaunSpZs+erT/96U+qqKgwnSXp1C8Oy5YtU5cuXc55LAMwAKCB/fv3Kzc3V8OGDZNVLhcfGxur\nkJAQSVJVVZUkKTDQ/O1Vr7766rqVzH79+ikmJkb79+83XCXFxMQoOTlZUVFRplNUWlqq9PR0CRyf\nJAAABD9JREFUTZs2TU6nU1dddZVyc3N16NAho10DBgzQRRddpA4dGt6lzCSn06nrr79ekZGRkqRL\nLrlEx44dU1FRkeEyqVu3bnI6nfJ6vaqurlZgYKBlfkl+7733NHz4cIWHh5/zWAZgAEA9Xq9XL774\nom6++WbLDL+npaWlafbs2br77rs1bdo09e3b13RSPcXFxTp69KgSEhJMp1jKsWPH5HQ6FRgYqN//\n/vc6fvy4YmJilJ2dbTqtTdizZ48iIyMVGhpqOkWS9Pe//12zZs3SokWLNH/+fPn7+5tOUkFBgVJS\nUnT11Vc36ngGYABAPWvXrlW3bt0UHx9vmZWd0yZMmKCXX35ZDzzwgJYvX64DBw6YTqpn8eLFSk5O\nVlxcnOkUS6moqFBgYKDKysp05MgRFRcXKygoSOXl5abTLK+0tFRLlizRzTffbDqlzu23366XXnpJ\nM2fO1KJFi1RZWWk6SS+99JKuvfZaOZ3ORh3Pm+AAwAe9/vrreuuttxo8P2DAAJ04cUKPPPKIJLX6\nCvD3dY0aNUrz5s2re5yYmKiLLrpIGzZsUPfu3S3RtWzZMpWUlOiuu+5q8Z6mdFlBQECAysvL5XK5\n9I9//EOSVFZWZoktLFZWVVWlJ554QuPHj9fYsWNN59TjcDh05ZVXavXq1dqxY4eGDx9urGX37t3K\nzc3VuHHjGv0aBmAA8EEzZszQjBkzGjx/4MABzZ8/Xz/5yU/qPZ+VlaWFCxca6zqT1hzOz9W1cuVK\nbd++Xffff78cDodluqyic+fOqqysVF5eniIjI1VdXa2cnBxWys+itrZWTz31lGJjYy39M7bCNqn9\n+/drz549mjlzZt1zO3fu1OHDh3XLLbec8TUMwACAOt27d9drr71W9/iNN95QTk6O7rjjDoNVp6xa\ntUpjxoxRRESEMjIy9Mknn+jee+81naX169fro48+0kMPPWS5Fc3KykrV1NRIOrWaaLPZ5OfX+n/1\nBwcHa8iQIVqxYoVmzZqlVatWKSoqyvhe6draWlVXV6u2tla1tbWqqqqSw+GQ3W5+h+jixYtls9l0\n++23m06pU1BQoK1bt2rMmDEKCAjQ2rVrdfLkSeN78adMmaIpU6bUPX7wwQd18cUXn/UyaAzAAIA2\n4dChQ3r33XdVUlKiyMhIzZo1S4MGDTKdpTfffFP5+fn1fkmYPn26pk2bZrBKOn78eL3rN8+aNUsD\nBgzQ/fffb6Tn9CXibr31VnXp0kV33323kY5vS01N1XPPPVf3eMOGDbrhhht0/fXXG6yScnNztW7d\nOvn7+2vOnDl1z993333q37+/sS673a60tDS98sorqq6uVnx8vObPn2+5q2g0hs1rhbVrAAAAoJWY\nX+MHAAAAWhEDMAAAAHwKAzAAAAB8CgMwAAAAfAoDMAAAAHwKAzAAAAB8CgMwAAAAfAoDMAAAAHwK\nAzAAAAB8yv8CxeybdoITQvwAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eeb7e0b8>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This result may be somewhat suprising to you. The transfer function looks \"fairly\" linear - it is pretty close to a straight line, but the probability distribution of the output is completely different from a Gaussian.  Recall the equations for multiplying two univariate Gaussians:\n",
      "$$\\begin{aligned}\n",
      "\\mu =\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}\\mbox{, } \n",
      "\\sigma = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "These equations do not hold for non-Gaussians, and certainly do not hold for the probability distribution shown in the 'output' chart above. \n",
      "\n",
      "Think of what this implies for the Kalman filter algorithm of the previous chapter. All of the equations assume that a Gaussian passed through the process function results in another Gaussian. If this is not true then all of the assumptions and guarantees of the Kalman filter do not hold. Let's look at what happens when we pass the output back through the function again, simulating the next step time step of the Kalman filter."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y=g(data)\n",
      "plot_transfer_func (y, g, lims=(-4,4), num_bins=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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A4VtEedgAE73i4NloAMCzZ8/UlkfGJKBdw2poWttBjFhERIWSyWRo4eWEFl5O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\n4VlJAuDj898AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb60cecea58>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As you can see the probability function is futher distorted from the original Gaussian. However, the graph is still somewhat symmetric around $0$, let's see what the mean is."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print ('input  mean, variance: %.4f, %.4f'% (np.average(data), np.std(data)**2))\n",
      "print ('output mean, variance: %.4f, %.4f'% (np.average(y), np.std(y)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "input  mean, variance: -0.0021, 1.0008\n",
        "output mean, variance: -0.0242, 2.2456\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's compare that to the linear function that passes through (-2,3) and (2,-3), which is very close to the nonlinear function we have plotted. Using the equation of a line we have\n",
      "$$m=\\frac{-3-3}{2-(-2)}=-1.5$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def h(x): return -1.5*x\n",
      "plot_transfer_func (data, h, lims=(-4,4), num_bins=300)\n",
      "out = h(data)\n",
      "print ('output mean, variance: %.4f, %.4f'% (np.average(out), np.std(out)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9T0lHl2auaOVul7dMJoOkTrgjIiqvhq5W2D+7J37Y/Re+3fknNhyJRvil+1g4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\nAACvwgAMAAAAr/K/0Ey3JPajtzQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5efa83f98>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "output mean, variance: 0.0032, 2.2518\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Although the shapes of the output are very different, the mean and variance of each are almost the same. This may lead us to reasoning that perhaps we can ignore this problem if the nonlinear equation is 'close to' linear. To test that, we can iterate several times and then compare the results."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "out = h(data)\n",
      "out2 = g(data)\n",
      "\n",
      "for i in range(10):\n",
      "    out = h(out)\n",
      "    out2 = g(out2)\n",
      "print ('linear    output mean, variance: %.4f, %.4f' % \n",
      "       (np.average(out), np.std(out)**2))\n",
      "print ('nonlinear output mean, variance: %.4f, %.4f' % \n",
      "       (np.average(out2), np.std(out2)**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "linear    output mean, variance: 0.1856, 7487.7480\n",
        "nonlinear output mean, variance: -1.4677, 26256.5001\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Unfortunately we can see that the nonlinear version is not stable. We have drifted significantly from the mean of 0, and the variance is half an order of magnitude larger. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The Extended Kalman Filter"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The extended Kalman filter (EKF) works by linearizing the system model at each update. For example, consider the problem of tracking a cannonball in flight. Obviously it follows a curved flight path. However, if our update rate is small enough, say 1/10 second, then the trajectory over that time is nearly linear. If we linearize that short segment we will get an answer very close to the actual value, and we can use that value to perform the prediction step of the filter. There are many ways to linearize a set of nonlinear differential equations, and the topic is somewhat beyond the scope of this book. In practice, a Taylor series approximation is frequently used with EKFs, and that is what we will use. \n",
      "\n",
      "\n",
      "Consider the function $f(x)=x^2\u22122x$, which we have plotted below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xs =  np.arange(0,2,0.01)\n",
      "ys = [x**2 - 2*x for x in xs]\n",
      "plt.plot (xs, ys)\n",
      "plt.xlim(1,2)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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pPKCCUaoBAKiCDMPQV+ti9eqXW5WdVyjfOs5686/9NLhroNnRgBqJUg0AQBVz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\nIjwBbNIAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5efbd2c18>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We want a linear appoximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation (update) of the Kalman filter we know its current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
      "\n",
      "We can use any linear function that passes through the curve at (1.5,-0.75). For example, consider using f(x)=8x\u221212.75 as the linearization, as in the plot below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def y(x): \n",
      "    return 8*x - 12.75\n",
      "plt.plot (xs, ys,c='k')\n",
      "plt.plot ([1.25, 1.75], [y(1.25), y(1.75)], c='r')\n",
      "plt.xlim(1,2)\n",
      "plt.ylim([-1.5, 1])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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z5+vVV1+1dgwAQGejk9pptb7BsDVEf/PNN2pubjb2R0REKDk5WVOnTlVKSop6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\nAAAAwEb/P5ViKQ6nyfMKAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eeb53710>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is not a good linearization for $f(x)$. It is exact for $x=1.5$, but quickly diverges when $x$ varies by a small amount.\n",
      "\n",
      "A much better approach is to use the slope of the function at the evaluation point as the linearization. We find the slope by taking the first derivative of the function:\n",
      "\n",
      " $$f(x) = x^2 -2x \\\\\n",
      " \\frac{df}{dx} = 2x - 2$$, \n",
      " \n",
      " so the slope at 1.5 is $2*1.5-2=1$. Let's plot that."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def y(x): \n",
      "    return x - 2.25\n",
      "\n",
      "plt.plot (xs, ys,c='k')\n",
      "plt.plot ([1,2], [y(1),y(2)], c='r')\n",
      "plt.xlim(1,2)\n",
      "plt.ylim([-1.5, 1])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9/vnnstfV6T5JcZLSJMV7e+trdvv1gJauR7TNRkAD6DFENQDAo1VVVemLL75Q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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eeb21160>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we can see that this linearization is much better. It is still exactly correct at $x=1.5$, but the errors are very small as x varies. Compare the tiny error at $x=1.4$ vs the very large error at $x=1.4$ in the previous plot. This does not constitute a formal proof of correctness, but this sort of geometric depiction should be fairly convincing. Certainly it is easy to see that in this case if the line had any other slope the errors would accumulate more quickly. \n",
      "\n",
      "To implement the extended Kalman filter we will leave the linear equations as they are, and use partial derivatives to evaluate the system matrix $\\mathbf{F}$ and the measurement matrix $\\mathbf{H}$ at the state at time t ($\\mathbf{x}_t$). Since $\\mathbf{F}$ also depends on the control input vector $\\mathbf{u}$ we will need to include that term:\n",
      "\n",
      "$$\n",
      "\\begin{aligned}\n",
      "F \n",
      "&\\equiv {\\frac{\\partial{f}}{\\partial{x}}}\\biggr|_{{x_t},{u_t}} \\\\\n",
      "H &\\equiv \\frac{\\partial{h}}{\\partial{x}}\\biggr|_{x_t} \n",
      "\\end{aligned}\n",
      "$$\n",
      "\n",
      "All this means is that at each update step we compute $\\mathbf{F}$ as the partial derivative of our function $f()$ evaluated at  x. \n",
      "\n",
      "We approximate the state transition function $\\mathbf{F}$ by using the Taylor-series expansion \n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "** orphan text\n",
      "This approach has many issues. First, of course, is the fact that the linearization does not produce an exact answer. More importantly, we are not linearizing the actual path, but our filter's estimation of the path. We linearize the estimation because it is statistically likely to be correct; but of course it is not required to be. So if the filter's output is bad that will cause us to linearize an incorrect estimate, which will almost certainly lead to an even worse estimate. In these cases the filter will quickly diverge. This is where the 'black art' of Kalman filter comes in. We are trying to linearize an estimate, and there is no guarantee that the filter will be stable. A vast amount of the literature on Kalman filters is devoted to this problem. Another issue is that we need to linearize the system using analytic methods. It may be difficult or impossible to find an analytic solution to some problems. In other cases we may be able to find the linearization, but the computation is very expensive. **\n",
      "\n",
      "In the next chapter we will spend a lot of time on a new development, the unscented Kalman filter(UKF) which avoids many of these problems. I think that as it becomes better known it will supplant the EKF in most applications, though that is still an open question. Certainly research has shown that the UKF performs at least as well as, and often much better than the EKF. \n",
      "\n",
      "I think the easiest way to understand the EKF is to just start off with an example. Perhaps the reason for some of my mathmatical choices will not be clear, but trust that the end result will be an EKF."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example: Tracking a Flying Airplane"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will start by simulating tracking an airplane by using ground based radar. Radars work by emitting a beam of radio waves and scanning for a return bounce. Anything in the beam's path will reflects some of the signal back to the radar. By timing how long it takes for the reflected signal to get back to the radar the system can compute the *slant distance* - the straight line distance from the radar installation to the object.\n",
      "\n",
      "For this example we want to take the slant range measurement from the radar and compute the horizontal position (distance of aircraft from the radar measured over the ground) and altitude of the aircraft, as in the diagram below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import ekf_internal\n",
      "ekf_internal.show_radar_chart()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GKwAVLeqWLQqeP1++n32mvIkTlTN9ukpvvNHVMwM8Rn5+vgzDUFhYmEpLS/Xy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\nCKsAAACwrP8Ps4Y77W7t8j0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eeb20860>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GKwAVLeqWLQqeP1++n32mvIkTlTN9ukpvvNHVMwM8Rn5+vgzDUFhYmEpLS/Xy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\nCKsAAACwrP8Ps4Y77W7t8j0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5efbe2128>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As discussed in the introduction, our measurement model is the nonlinear function $x=\\sqrt{slant^2 - altitude^2}$. Therefore we will need a nonlinear \n",
      "\n",
      "Predict step:\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "\\textbf{Linear} & \\textbf{Nonlinear} \\\\\n",
      "x = Fx & x = \\underline{f(x)} \\\\\n",
      "P = FPF^T + Q & P = FPF^T + Q\n",
      "\\end{array}\n",
      "$$\n",
      "\n",
      "Update step:\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "\\textbf{Linear} & \\textbf{Nonlinear} \\\\\n",
      "K = PH^T(HPH^T + R)^{-1}& K = PH^T(HPH^T + R)^{-1}\\\\\n",
      "x = x + K(z-Hx) & x = x + K(z-\\underline{h(x)}) \\\\\n",
      "P = P(I - KH) & P = P(I - KH)\\\\\n",
      "\\end{array}\n",
      "$$\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As we can see there are two minor changes to the Kalman filter equations, which I have underlined. The first change replaces the equation $\\mathbf{x} = \\mathbf{Fx}$ with $\\mathbf{x} = f(\\mathbf{x})$. In the Kalman filter, $\\mathbf{Fx}$ is how we compute the new state based on the old state. However, in a nonlinear system we cannot use linear algebra to compute this transition. So instead we hypothesize a nonlinear function $f()$ which performs this function. Likewise, in the Kalman filter we convert the state to a measurement with the linear function $\\mathbf{Hx}$. For the extended Kalman filter we replace this with a nonlinear function $h()$, giving $\\mathbf{z}_x = h(\\mathbf{x})$.\n",
      "\n",
      "The only question left is how do we implement use $f()$ and $h()$ in the Kalman filter if they are nonlinear? We reach for the single tool that we have available for solving nonlinear equations - we linearize them at the point we want to evaluate the system.  For example, consider the function $f(x) = x^2 -2x$\n",
      "\n",
      "\n",
      "The rest of the equations are unchanged, so $f()$ and $h()$ must produce a matrix that approximates the values of the matrices $\\mathbf{F}$ and $\\mathbf{H}$ at the current value for $\\mathbf{x}$. We do this by computing the partial derivatives of the state and measurements functions:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the State Variables\n",
      "\n",
      "So we want to track the position of an aircraft assuming a constant velocity and altitude, and measurements of the slant distance to the aircraft. That means we need 3 state variables - horizontal distance, velocity, and altitude.\n",
      "\n",
      "$$\\mathbf{x} = \\begin{bmatrix}distance \\\\velocity\\\\ altitude\\end{bmatrix}=  \\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the System Model\n",
      "We will model this as a set of differential equations. So we need an equation in the form \n",
      "$$\\dot{\\mathbf{x}} = \\mathbf{Ax} + \\mathbf{w}$$\n",
      "\n",
      "where $\\mathbf{w}$ is the system noise. \n",
      "\n",
      "Let's work out the equation for each of the rows in $\\mathbf{x}.$\n",
      "\n",
      "The first row is $\\dot{x}_{pos}$, which is the velocity of the airplane. So we can say \n",
      "\n",
      "$$\\dot{x}_{pos} = x_{vel}$$\n",
      "\n",
      "The second row is $\\dot{x}_{vel}$, which is the acceleration of the airplane. We assume constant velocity, so the acceleration equals zero. However, we also assume system noise due to things like buffeting winds, errors in control inputs, and so on, so we need to add an error $w_{acc}$ to the term, like so\n",
      "\n",
      "$$\\dot{x}_{vel} = 0 + w_{acc}$$\n",
      "\n",
      "\n",
      "The final row contains $\\dot{x}_{alt}$, which is the rate of change in the altitude. We assume a constant altitude, so this term is 0, but as with acceleration we need to add in a noise term to account for things like wind, air density, and so on. This gives us\n",
      "\n",
      "$$\\dot{x}_{alt} = 0 + w_{alt}$$\n",
      "\n",
      "We turn this into matrix form with the following:\n",
      "\n",
      "$$\\dot{\\mathbf{x}} = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0& 0& 0 \\\\ 0&0&0\\end{bmatrix}\n",
      "\\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix} + \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "Now we have our differential equations for the system we can somehow solve for them to get our familiar Kalman filter state equation\n",
      "\n",
      "$$ \\mathbf{x}=\\mathbf{Fx}$$\n",
      "\n",
      "Solving an arbitrary set of differential equations is beyond the scope of this book, however most Kalman filters are amenable to Taylor-series expansion which I will briefly explain here without proof. \n",
      "\n",
      "Given the partial differential equation \n",
      "\n",
      "$$\\mathbf{F} = \\frac{\\partial f(\\mathbf{x})}{\\partial x}$$\n",
      "\n",
      "the solution is $e^{\\mathbf{F}t}$. This is a standard answer learned in a first year partial differential equations course, and is not intuitively obvious from the material presented so far. However, we can compute the exponential matrix $e^{\\mathbf{F}t}$ using a Taylor-series expansion in the form:\n",
      "\n",
      "$$\\Phi = \\mathbf{I} + \\mathbf{F}\\Delta t + \\frac{(\\mathbf{F}\\Delta t)^2}{2!} +  \\frac{(\\mathbf{F}\\Delta t)^3}{3!} + \\ldots$$\n",
      "\n",
      "You may expand that equation to as many terms as required for accuracy, however many problems only use the first term\n",
      "\n",
      "$$\\Phi \\approx \\mathbf{I} + \\mathbf{F}\\Delta t$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\Phi$ is our system matrix. We cannot use greek symbols in Python, so the code uses the symbol `F` for $\\Phi$. This is admittedly confusing. In the math above $\\mathbf{F}$ represents the system of partial differential equations, and $\\Phi$ is the system matrix. In the Python the partial differential equations are not represented in the code, and the system matrix is `F`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Design the Measurement Model\n",
      "\n",
      "The measurement function for our filter needs to take the filter state $\\mathbf{x}$ and turn it into a slant range distance. This is nothing more than the pythagorean theorem.\n",
      "\n",
      "$$h(\\mathbf{x}) = \\sqrt{x_{pos}^2 + x_{alt}^2}$$\n",
      "\n",
      "\n",
      "The relationship between the slant distance and the position on the ground is nonlinear due to the square root term.\n",
      "So what we need to do is linearize the measurement function at some point. As we discussed above, the best way to linearize an equation at a point is to find its slope, which we do by taking its derivative.\n",
      "\n",
      "$$\n",
      "\\mathbf{H} \\equiv \\frac{\\partial{h}}{\\partial{x}}\\biggr|_x \n",
      "$$\n",
      "\n",
      "The derivative of a matrix is called a Jacobian, which in general takes the form \n",
      "\n",
      "$$\\frac{\\partial \\mathbf{h}}{\\partial \\mathbf{x}} = \n",
      "\\begin{bmatrix}\n",
      "\\frac{\\partial h_1}{\\partial x_1} & \\frac{\\partial h_1}{\\partial x_2} &\\dots \\\\\n",
      "\\frac{\\partial h_2}{\\partial x_1} & \\frac{\\partial h_2}{\\partial x_2} &\\dots \\\\\n",
      "\\vdots & \\vdots\n",
      "\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "In other words, each element in the matrix is the partial derivative of the function $h$ with respect to the variables $x$. For our problem we have\n",
      "\n",
      "$$\\mathbf{H} = \\begin{bmatrix}\\frac{\\partial h}{\\partial x_{pos}} & \\frac{\\partial h}{\\partial x_{vel}} & \\frac{\\partial h}{\\partial x_{alt}}\\end{bmatrix}$$\n",
      "\n",
      "where $h(x) = \\sqrt{x_{pos}^2 + x_{alt}^2}$ as given above.\n",
      "\n",
      "Solving each in turn:\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{pos}} &= \\\\ &=\\frac{\\partial}{\\partial x_{pos}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ &= \\frac{x_{pos}}{\\sqrt{x^2 + x_{alt}^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "and\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{vel}} &=\\\\\n",
      "&= \\frac{\\partial}{\\partial x_{vel}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ \n",
      "&= 0\n",
      "\\end{aligned}$$\n",
      "\n",
      "and\n",
      "$$\\begin{aligned}\n",
      "\\frac{\\partial h}{\\partial x_{alt}} &=\\\\ &= \\frac{\\partial}{\\partial x_{alt}} \\sqrt{x_{pos}^2 + x_{alt}^2} \\\\ &= \\frac{x_{alt}}{\\sqrt{x_{pos}^2 + x_{alt}^2}}\n",
      "\\end{aligned}$$\n",
      "\n",
      "giving us \n",
      "\n",
      "$$\\mathbf{H} = \n",
      "\\begin{bmatrix}\n",
      "\\frac{x_{pos}}{\\sqrt{x_{pos}^2 + x_{alt}^2}} & \n",
      "0 &\n",
      "&\n",
      "\\frac{x_{alt}}{\\sqrt{x_{pos}^2 + x_{alt}^2}}\n",
      "\\end{bmatrix}$$\n",
      "\n",
      "This may seem daunting, so step back and recognize that all of this math is just doing something very simple. We have an equation for the slant range to the airplane which is nonlinear. The Kalman filter only works with linear equations, so we need to find a linear equation that approximates $\\mathbf{H}$ As we discussed above, finding the slope of a nonlinear equation at a given point is a good approximation. For the Kalman filter, the 'given point' is the state variable $\\mathbf{x}$ so we need to take the derivative of the slant range with respect to $\\mathbf{x}$. \n",
      "\n",
      "To make this more concrete, let's now write a Python function that computes the Jacobian of $\\mathbf{H}$. The `ExtendedKalmanFilter` class will be using this to generate `ExtendedKalmanFilter.H` at each step of the process."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import sqrt\n",
      "def HJacobian_at(x):\n",
      "    \"\"\" compute Jacobian of H matrix for state x \"\"\"\n",
      "\n",
      "    horiz_dist = x[0]\n",
      "    altitude   = x[2]\n",
      "    denom = sqrt(horiz_dist**2 + altitude**2)\n",
      "    return array ([[horiz_dist/denom, 0., altitude/denom]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, let's provide the code for $h(\\mathbf{x})$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def hx(x):\n",
      "    \"\"\" compute measurement for slant range that would correspond \n",
      "    to state x.\n",
      "    \"\"\"\n",
      "    \n",
      "    return (x[0]**2 + x[2]**2) ** 0.5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now lets write a simulation for our radar."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy.random import randn\n",
      "import math\n",
      "\n",
      "class RadarSim(object):\n",
      "    \"\"\" Simulates the radar signal returns from an object flying \n",
      "    at a constant altityude and velocity in 1D. \n",
      "    \"\"\"\n",
      "    \n",
      "    def __init__(self, dt, pos, vel, alt):\n",
      "        self.pos = pos\n",
      "        self.vel = vel\n",
      "        self.alt = alt\n",
      "        self.dt = dt\n",
      "        \n",
      "    def get_range(self):\n",
      "        \"\"\" Returns slant range to the object. Call once for each\n",
      "        new measurement at dt time from last call.\n",
      "        \"\"\"\n",
      "        \n",
      "        # add some process noise to the system\n",
      "        vel = self.vel  + 5*randn()\n",
      "        alt = self.alt + 10*randn()\n",
      "        self.pos = self.pos + vel*self.dt\n",
      "    \n",
      "        # add measurment noise\n",
      "        err = self.pos * 0.05*randn()\n",
      "        slant_dist = math.sqrt(self.pos**2 + alt**2)\n",
      "        \n",
      "        return slant_dist + err"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can implement our filter. I have not yet designed $\\mathbf{R}$ and $\\mathbf{Q}$ which is required to get optimal performance. However, we have already covered a lot of confusing material and I want you to see concrete examples as soon as possible. Therefore I will use 'reasonable' values for $\\mathbf{R}$ and $\\mathbf{Q}$.\n",
      "\n",
      "The `FilterPy` library provides the class `ExtendedKalmanFilter`. It works very similar to the `KalmanFilter` class we have been using, except that it allows you to provide functions that compute the Jacobian of $\\mathbf{H}$ and the function $h(\\mathbf{x})$. We have already written the code for these two functions, so let's just get going.\n",
      "\n",
      "We start by importing the filter and creating it. There are 3 variables in `x` and only 1 measurement. At the same time we will create our radar simulator.\n",
      "\n",
      "    from filterpy.kalman import ExtendedKalmanFilter\n",
      "\n",
      "    rk = ExtendedKalmanFilter(dim_x=3, dim_z=1)\n",
      "    radar = RadarSim(dt, pos=0., vel=100., alt=1000.)\n",
      "    \n",
      "We will initialize the filter near the airplane's actual position\n",
      "\n",
      "    rk.x = array([radar.pos, radar.vel-10, radar.alt+100])\n",
      "    \n",
      "We assign the system matrix using the first term of the Taylor series expansion we computed above.\n",
      "\n",
      "    dt = 0.05\n",
      "    rk.F = eye(3) + array ([[0, 1, 0],\n",
      "                            [0, 0, 0],\n",
      "                            [0, 0, 0]])*dt\n",
      "                            \n",
      "After assigning reasonble values to $\\mathbf{R}$, $\\mathbf{Q}$, and $\\mathbf{P}$ we can run the filter with a simple loop\n",
      "\n",
      "    for i in range(int(20/dt)):\n",
      "        z = radar.get_range()\n",
      "        rk.update(array([[z]]), HJacobian_at, hx)\n",
      "        rk.predict()\n",
      "                       \n",
      "Putting that all together along with some boilerplate code to save the results and plot them, we get"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from filterpy.kalman import ExtendedKalmanFilter\n",
      "from numpy import eye, array, asarray\n",
      "\n",
      "dt = 0.05\n",
      "rk = ExtendedKalmanFilter(dim_x=3, dim_z=1)\n",
      "radar = RadarSim(dt, pos=0., vel=100., alt=1000.)\n",
      "\n",
      "rk.x = array([radar.pos, radar.vel+100, radar.alt+1000])\n",
      "\n",
      "\n",
      "rk.F = eye(3) + array ([[0, 1, 0],\n",
      "                        [0, 0, 0],\n",
      "                        [0, 0, 0]])*dt\n",
      "\n",
      "rk.R = radar.alt * 0.05 # 5% of distance\n",
      "rk.Q = array([[0, 0, 0],\n",
      "              [0, 1, 0],\n",
      "              [0, 0, 1]]) * 0.001\n",
      "\n",
      "rk.P *= 50\n",
      "\n",
      "\n",
      "xs = []\n",
      "for i in range(int(20/dt)):\n",
      "    z = radar.get_range()\n",
      "    \n",
      "    rk.update(array([z]), HJacobian_at, hx)\n",
      "    xs.append(rk.x)\n",
      "    rk.predict()\n",
      "\n",
      "\n",
      "xs = asarray(xs)\n",
      "\n",
      "plt.figure()\n",
      "plt.plot(xs[:,0])\n",
      "plt.ylabel('position')\n",
      "\n",
      "plt.figure()\n",
      "plt.plot(xs[:,1])\n",
      "plt.ylabel('velocity')\n",
      "\n",
      "plt.figure()\n",
      "plt.plot(xs[:,2])\n",
      "plt.ylabel('altitude')\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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XERERkStSW9/Ai6vSeOvLw1yorG00F9LFkzkTBnPHmAgcHbQjyy+hgC4iIiIiP6usooaP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\nhfw/XRxKdOiGnP0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eebfcf28>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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IXbAohN911114eHjw7bffVm+U4+XlxejRo7nlllvqtUB71q1di8oQfvyMQriI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\n7Y2KYjq11K7jys3RVid7fYI/c5cd5B+rD/LPNQe5ZeyTFrb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       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eea7c898>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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4h/PzzZwP+shHPqJPfepTevrpp/We97xHPp9PwWBQa9eulSRt27ZNfr9fkjL76urqVFdX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\nJyIiIiJyiD+dDQ41Kpgc8QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb5eea48b38>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Using SymPy to compute Jacobians\n",
      "\n",
      "Depending on your experience with derivatives you may have found the computation of the Jacobian above either fairly straightforward, or quite difficult. Even if you found it easy, a slightly more difficult problem easily leads to very difficult computations.\n",
      "\n",
      "As explained in Appendix A, we can use the SymPy package to compute the Jacobian for us. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "sympy.init_printing(use_latex='mathjax')\n",
      "\n",
      "x_pos, x_vel, x_alt = sympy.symbols('x_pos, x_vel x_alt')\n",
      "\n",
      "H = sympy.Matrix([sympy.sqrt(x_pos**2 + x_alt**2)])\n",
      "\n",
      "state = sympy.Matrix([x_pos, x_vel, x_alt])\n",
      "H.jacobian(state)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{x_{pos}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}} & 0 & \\frac{x_{alt}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "\u23a1       x_pos                    x_alt        \u23a4\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  0  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2   _________________        _________________\u23a5\n",
        "\u23a2  \u2571      2        2        \u2571      2        2 \u23a5\n",
        "\u23a3\u2572\u2571  x_alt  + x_pos       \u2572\u2571  x_alt  + x_pos  \u23a6"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This result is the same as the result we computed above, and at much less effort on our part!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Designing Q\n",
      "\n",
      "Now we need to design the process noise matrix $\\mathbf{Q}$. From the previous section we have the system equation\n",
      "\n",
      "$$\\dot{\\mathbf{x}} = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0& 0& 0 \\\\ 0&0&0\\end{bmatrix}\n",
      "\\begin{bmatrix}x_{pos} \\\\x_{vel}\\\\ x_{alt}\\end{bmatrix} + \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}\n",
      "$$\n",
      "\n",
      "where our process noise is\n",
      "\n",
      "$$w = \\begin{bmatrix}0 \\\\w_{vel}\\\\ w_{alt}\\end{bmatrix}$$\n",
      "\n",
      "We know from the Kalman filter math chapter that \n",
      "\n",
      "$$\\mathbf{Q} = E(ww^T)$$\n",
      "\n",
      "where $E(\\bullet)$ is the expected value. We compute the expected value as\n",
      "\n",
      "$$\\mathbf{Q} = \\int_0^{dt} \\Phi(t)\\mathbf{Q}\\Phi^T(t) dt$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Rather than do this by hand, let's use sympy."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "sympy.init_printing(use_latex='mathjax')\n",
      "w_vel, w_alt, dt = sympy.symbols('w_vel w_alt \\Delta{t}')\n",
      "w = sympy.Matrix([[0, w_vel, w_alt]]).T\n",
      "phi = sympy.Matrix([[1, dt, 0], [0, 1, 0], [0,0,1]])\n",
      "\n",
      "q = w*w.T\n",
      "\n",
      "sympy.integrate(phi*q*phi.T, (dt, 0, dt))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{\\Delta{t}^{3} w_{vel}^{2}}{3} & \\frac{\\Delta{t}^{2} w_{vel}^{2}}{2} & \\frac{w_{alt} w_{vel}}{2} \\Delta{t}^{2}\\\\\\frac{\\Delta{t}^{2} w_{vel}^{2}}{2} & \\Delta{t} w_{vel}^{2} & \\Delta{t} w_{alt} w_{vel}\\\\\\frac{w_{alt} w_{vel}}{2} \\Delta{t}^{2} & \\Delta{t} w_{alt} w_{vel} & \\Delta{t} w_{alt}^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "\u23a1           3      2                2      2             2            \u23a4\n",
        "\u23a2  \\Delta{t} \u22c5w_vel        \\Delta{t} \u22c5w_vel     \\Delta{t} \u22c5w_alt\u22c5w_vel\u23a5\n",
        "\u23a2  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2          3                       2                      2           \u23a5\n",
        "\u23a2                                                                     \u23a5\n",
        "\u23a2           2      2                                                  \u23a5\n",
        "\u23a2  \\Delta{t} \u22c5w_vel                       2                           \u23a5\n",
        "\u23a2  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       \\Delta{t}\u22c5w_vel      \\Delta{t}\u22c5w_alt\u22c5w_vel \u23a5\n",
        "\u23a2          2                                                          \u23a5\n",
        "\u23a2                                                                     \u23a5\n",
        "\u23a2         2                                                           \u23a5\n",
        "\u23a2\\Delta{t} \u22c5w_alt\u22c5w_vel                                           2   \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \\Delta{t}\u22c5w_alt\u22c5w_vel     \\Delta{t}\u22c5w_alt    \u23a5\n",
        "\u23a3          2                                                          \u23a6"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example: A falling Ball"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the **Designing Kalman Filters** chapter I first considered tracking a ball in a vacuum, and then in the atmosphere. The Kalman filter performed very well for vacuum, but diverged from the ball's path in the atmosphere. Let us look at the output; to avoid littering this chapter with code from that chapter I have placed it all in the file `ekf_internal.py'."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import ekf_internal\n",
      "ekf_internal.plot_ball()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9uq67IIGxEEL8GaiysjBs3Yrhm2/Qb9uGuporCNgCAkoD4Utf9gYNXNzTm4xG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zy9UFpaqo0bN6q0tFTdu3fXrFmzNGrUKLK8QYcPH9bu3bt15swZjRkzRosXL5akv8yP6/q12sqS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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fb60cecea20>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can artifically force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
      "\n",
      "Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
      "\n",
      "$$a_x = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_x \\\\\n",
      "a_y = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_y- g\n",
      "$$\n",
      "\n",
      "These equations will be very unpleasant to work with while we develop this subject, so for now I will retreat to a simpler one dimensional problem using this simplified equation for acceleration that does not take the nonlinearity of the drag coefficient into account:\n",
      "\n",
      "\n",
      "$$\\ddot{x} = \\frac{0.0034ge^{-x/20000}\\dot{x}^2}{2\\beta} - g$$\n",
      "\n",
      "Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    }
   ],
   "metadata": {}
  }
 ]
}