Added warnings about unfinished EKF chapter.

The EKF chapter is a very early draft/dump of material, and
yet people are trying to use it. They really shouldn't.

09_Extended_Kalman_Filters/Extended_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -261,7 +261,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f6bf4f16128>"
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@@ -271,7 +271,7 @@
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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",
+      "**Author's note: this is still being heavily edited and added to - there is a lot of duplicate material, incorrect code, and so on. The examples are just a brain dump, not fully formulated and not ready to be used.**\n",
       "\n",
       "The 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",
@@ -354,9 +354,9 @@
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@@ -386,9 +386,9 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd8aeceb8>"
+        "<matplotlib.figure.Figure at 0x7f4bacf25f28>"
        ]
       }
      ],
@@ -420,9 +420,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd83223c8>"
+        "<matplotlib.figure.Figure at 0x7f4bac74ae80>"
        ]
       }
      ],
@@ -456,9 +456,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bf4f05dd8>"
+        "<matplotlib.figure.Figure at 0x7f4bc93e7c18>"
        ]
       }
      ],
@@ -485,8 +485,8 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "input  mean, variance: 0.0003, 0.9969\n",
-        "output mean, variance: -0.0283, 2.2320\n"
+        "input  mean, variance: -0.0004, 1.0027\n",
+        "output mean, variance: -0.0268, 2.2466\n"
        ]
       }
      ],
@@ -515,16 +515,16 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd8250128>"
+        "<matplotlib.figure.Figure at 0x7f4bac6b50f0>"
        ]
       },
       {
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "output mean, variance: -0.0005, 2.2430\n"
+        "output mean, variance: 0.0006, 2.2561\n"
        ]
       }
      ],
@@ -559,8 +559,8 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "linear    output mean, variance: -0.0296, 7458.6412\n",
-        "nonlinear output mean, variance: -1.9540, 26056.8570\n"
+        "linear    output mean, variance: 0.0360, 7502.2648\n",
+        "nonlinear output mean, variance: -1.7522, 26273.4649\n"
        ]
       }
      ],
@@ -609,7 +609,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd82813c8>"
+        "<matplotlib.figure.Figure at 0x7f4bac8c25c0>"
        ]
       }
      ],
@@ -644,7 +644,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd8254588>"
+        "<matplotlib.figure.Figure at 0x7f4bac68e438>"
        ]
       }
      ],
@@ -685,7 +685,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd827fd68>"
+        "<matplotlib.figure.Figure at 0x7f4bac712cc0>"
        ]
       }
      ],
@@ -756,7 +756,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd83bd2e8>"
+        "<matplotlib.figure.Figure at 0x7f4bacf18eb8>"
        ]
       }
      ],
@@ -1009,13 +1009,13 @@
       "        \"\"\"\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",
+      "        self.vel = self.vel  + .1*randn()\n",
+      "        self.alt = self.alt + .1*randn()\n",
+      "        self.pos = self.pos + self.vel*self.dt\n",
       "    \n",
       "        # add measurment noise\n",
       "        err = self.pos * 0.05*randn()\n",
-      "        slant_dist = math.sqrt(self.pos**2 + alt**2)\n",
+      "        slant_dist = math.sqrt(self.pos**2 + self.alt**2)\n",
       "        \n",
       "        return slant_dist + err"
      ],
@@ -1071,7 +1071,8 @@
       "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",
+      "# make an imperfect starting guess\n",
+      "rk.x = array([radar.pos-100, radar.vel+100, radar.alt+1000])\n",
       "\n",
       "\n",
       "rk.F = eye(3) + array ([[0, 1, 0],\n",
@@ -1082,13 +1083,20 @@
       "rk.Q = array([[0, 0, 0],\n",
       "              [0, 1, 0],\n",
       "              [0, 0, 1]]) * 0.001\n",
-      "\n",
+      "'''\n",
+      "wv = .1**2\n",
+      "wa = .1**2\n",
+      "rk.Q = array([[dt**3 * wv/3, dt**2*wv/2, 0],\n",
+      "              [dt**2*wv/2, dt*wv, 0],\n",
+      "              [0, 0, wa*dt]])'''\n",
       "rk.P *= 50\n",
       "\n",
       "\n",
       "xs = []\n",
+      "track = []\n",
       "for i in range(int(20/dt)):\n",
       "    z = radar.get_range()\n",
+      "    track.append((radar.pos, radar.vel, radar.alt))\n",
       "    \n",
       "    rk.update(array([z]), HJacobian_at, hx)\n",
       "    xs.append(rk.x)\n",
@@ -1096,18 +1104,31 @@
       "\n",
       "\n",
       "xs = asarray(xs)\n",
+      "track = asarray(track)\n",
+      "time = np.arange(0,len(xs)*dt, dt)\n",
       "\n",
       "plt.figure()\n",
-      "plt.plot(xs[:,0])\n",
-      "plt.ylabel('position')\n",
+      "plt.plot(time, track[:,0], label='track')\n",
+      "plt.plot(time, xs[:,0], label='filter')\n",
+      "plt.legend(loc=4)\n",
+      "plt.xlabel('time (sec)')\n",
+      "plt.ylabel('position (m)')\n",
+      "\n",
       "\n",
       "plt.figure()\n",
-      "plt.plot(xs[:,1])\n",
-      "plt.ylabel('velocity')\n",
+      "plt.plot(time, track[:,1], label='track')\n",
+      "plt.plot(time, xs[:,1])\n",
+      "plt.legend(loc=4)\n",
+      "plt.xlabel('time (sec)')\n",
+      "plt.ylabel('velocity (m/s)')\n",
       "\n",
       "plt.figure()\n",
-      "plt.plot(xs[:,2])\n",
-      "plt.ylabel('altitude')\n",
+      "plt.plot(time, track[:,2], label='track')\n",
+      "plt.plot(time, xs[:,2])\n",
+      "plt.ylabel('altitude (m)')\n",
+      "plt.legend(loc=4)\n",
+      "plt.xlabel('time (sec)')\n",
+      "plt.ylim((900,1600))\n",
       "plt.show()"
      ],
      "language": "python",
@@ -1116,25 +1137,25 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd409a7f0>"
+        "<matplotlib.figure.Figure at 0x7f4bac676048>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd41698d0>"
+        "<matplotlib.figure.Figure at 0x7f4bab812e10>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd7a55d68>"
+        "<matplotlib.figure.Figure at 0x7f4bab7e3b70>"
        ]
       }
      ],
@@ -1211,6 +1232,8 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "**author's note: ignore this, it  to be revised - noise in position and altitude is independent, not dependent**\n",
+      "\n",
       "Now we need to design the process noise matrix $\\mathbf{Q}$. From the previous section we have the system equation\n",
       "\n",
       "$$\\dot{\\mathbf{x}} = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0& 0& 0 \\\\ 0&0&0\\end{bmatrix}\n",
@@ -1309,13 +1332,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAyAAAAGNCAYAAADpbRVxAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f6bd8316d68>"
+        "<matplotlib.figure.Figure at 0x7f4ba85b1f98>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1344,7 +1367,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 20
     }
    ],
    "metadata": {}

exp/Untitled1.ipynb

@@ -0,0 +1,179 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:fc6923e502800422d1ca66394e71d567efb82c4dcd09c273561149f8d3b00fe5"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "Complementary Filter"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Convert continuous transfer function into discrete\n",
+      "\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import scipy.signal as sig\n",
+      "from math import sqrt\n",
+      "\n",
+      "dt = 0.01\n",
+      "ki = 0.1**2\n",
+      "kp = sqrt(2)*0.1\n",
+      "\n",
+      "num = [kp, ki]\n",
+      "den = [1, 0.]\n",
+      "sig.cont2discrete(sys=(num,den), dt=0.1)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 8,
+       "text": [
+        "(array([[ 0.14142136, -0.14042136]]), array([ 1., -1.]), 0.1)"
+       ]
+      }
+     ],
+     "prompt_number": 8
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "a = 5\n",
+      "num = [a, 0]\n",
+      "den = [a, 1]\n",
+      "sig.cont2discrete(sys=(num,den), dt=0.1)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 11,
+       "text": [
+        "(array([[ 1., -1.]]), array([ 1.        , -0.98019867]), 0.1)"
+       ]
+      }
+     ],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from sympy import *\n",
+      "from sympy.abc import *\n",
+      "a = Symbol('a', positive=True)\n",
+      "foo = (s/(s+a)).rewrite()\n",
+      "print(foo)\n",
+      "inverse_laplace_transform(1/((s+a)), s, t)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "s/(a + s)\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 39,
+       "text": [
+        "exp(-a*t)*Heaviside(t)"
+       ]
+      }
+     ],
+     "prompt_number": 39
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

exp/compute_q.ipynb

@@ -0,0 +1,54 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:72ca7769e63a40b2f7f89ae8f816dcd0df3658bb19519d8c95b2f6a5e99413ec"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from sympy import Symbol, Matrix\n",
+      "from sympy.interactive import printing\n",
+      "printing.init_printing()\n",
+      "\n",
+      "dt = Symbol('\\Delta t')\n",
+      "\n",
+      "Q = Matrix([[.5*dt**2], [.5*dt**2], [dt], [dt]])\n",
+      "Q*Q.T\n",
+      "\n",
+      "\n",
+      "Q = Matrix([[.5*dt**2], [dt]])\n",
+      "Q*Q.T"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "latex": [
+        "$$\\left[\\begin{matrix}0.25 \\Delta t^{4} & 0.5 \\Delta t^{3}\\\\0.5 \\Delta t^{3} & \\Delta t^{2}\\end{matrix}\\right]$$"
+       ],
+       "metadata": {},
+       "output_type": "pyout",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAKQAAAAzBAMAAADr+L6JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nD+iXYj8NWuVEAAAAAElFTkSuQmCC\n",
+       "prompt_number": 8,
+       "text": [
+        "\u23a1             4              3\u23a4\n",
+        "\u23a20.25\u22c5\\Delta t   0.5\u22c5\\Delta t \u23a5\n",
+        "\u23a2                             \u23a5\n",
+        "\u23a2            3             2  \u23a5\n",
+        "\u23a30.5\u22c5\\Delta t      \\Delta t   \u23a6"
+       ]
+      }
+     ],
+     "prompt_number": 8
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

exp/satellite.ipynb

@@ -0,0 +1,100 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:9fb5a469b0b0f27f1e721735fae76d74f0ad242621b94c3f528e9392b7deab82"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import math\n",
+      "\n",
+      "def range_to(x1, x2):\n",
+      "    dx = x1[0] - x2[0]\n",
+      "    dy = x1[1] - x2[1]\n",
+      "    print('dx',dx,dy)\n",
+      "    return math.sqrt(dx*dx + dy*dy)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "s1 = (0., 2.) # satellite 1\n",
+      "s2 = (1., 2.) # satellite 2\n",
+      "p = (-0.5, 1.) # platform postion\n",
+      "\n",
+      "\n",
+      "s1 = (0., 200.) # satellite 1\n",
+      "s2 = (100., 220.) # satellite 2\n",
+      "p = (10, 1.) # platform postion\n",
+      "\n",
+      "r1 = range_to(s1, p)\n",
+      "r2 = range_to(s2, p)\n",
+      "\n",
+      "x1, y1 = s1[0], s1[1]\n",
+      "x2, y2 = s2[0], s2[1]\n",
+      "\n",
+      "A = -(y2 - y1) / (x2 - x1)\n",
+      "B = (r1**2 - r2**2 - x1**2 - y1**2 + x2**2 + y2**2) / (2*(x2 - x1))\n",
+      "\n",
+      "a = 1 + A**2\n",
+      "b = -2*A*x1 + 2*A*B - 2*y1\n",
+      "c = x1**2 - 2*x1*B + y1**2 - r1**2\n",
+      "\n",
+      "y = (-b - math.sqrt(b**2 - 4*a*c)) / (2*a)\n",
+      "x = A*y + B\n",
+      "\n",
+      "print('p', x,y)\n",
+      "print('err', x-p[0], y-p[1])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "dx -10.0 199.0\n",
+        "dx 90.0 219.0\n",
+        "p 10.051726583768032 0.7413670811596572\n",
+        "err 0.05172658376803163 -0.2586329188403428\n"
+       ]
+      }
+     ],
+     "prompt_number": 55
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "A"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 56,
+       "text": [
+        "-0.2"
+       ]
+      }
+     ],
+     "prompt_number": 56
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}